Introduction to Econometric Production Analysis with R

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Introduction to Econometric Production Analysis with R (Fifth Draft Version) Arne Henningsen with contributions of Tomasz G. Czekaj

Department of Food and Resource Economics University of Copenhagen October 28, 2020

Foreword This is a collection of my lecture notes for various courses in the field of econometric production analysis. These lecture notes are still incomplete and may contain many typos, errors, and inconsistencies. Please report any problems to [email protected]. I am grateful to Tomasz G. Czekaj who drafted and revised some parts of these lecture notes and to my former students who helped me to improve my teaching and these notes through their questions, suggestions, and comments. Finally, I thank the R community for providing so many excellent tools for econometric production analysis. October 28, 2020 Arne Henningsen

How to cite these lecture notes: Henningsen, Arne (2020): Introduction to Econometric Production Analysis with R. Collection of Lecture Notes. 5th Draft Version. Department of Food and Resource Economics, University of Copenhagen. Available at http://leanpub.com/ProdEconR/.

2

History First Draft Version (March 9, 2015) ˆ Initial release at Leanpub.com

Second Draft Version (February 2, 2018) ˆ added a chapter on distance functions ˆ corrected the equation for calculating direct elasticities of substitution ˆ a large number of further additions, improvements, and corrections

Third Draft Version (January 29, 2019) ˆ corrected several typos and minor errors ˆ some minor additions and improvements

Fourth Draft Version (May 27, 2019) ˆ ˆ ˆ ˆ

added added added added

three sections about homotheticity of cost functions (sections 3.1.4, 3.2.6, and 3.5.6) a section about cost functions with multiple outputs (section 3.1.6) a section about cost functions with technical change (section 9.4) a proof that cost minimisation and profit maximisation imply that the output elas-

ticities of the inputs divided by the elasticity of scale are equal to their cost shares (section 2.1.11) ˆ added a proof that revenue maximisation and profit maximisation imply that the distance elasticities of the outputs derived from an output distance function are equal to their revenue shares (section 8.1.1.4) ˆ added a proof that cost minimisation and profit maximisation imply that the negative distance elasticities of the inputs divided by the elasticity of scale derived from an output distance function are equal to their cost shares (section 8.1.1.4) ˆ added an approach for obtaining unobserved (shadow) prices of outputs from an output distance function (section 8.1.1.5) ˆ a few minor additions and improvements Fifth Draft Version (October 28, 2020) ˆ added two subsections that discuss the suitability of the production function and the cost

function, respectively, in econometric applications (sections 2.1.13 and 3.1.7) ˆ added a comparison of the observed cost shares with the cost shares that would minimize costs according to an estimated Cobb-Douglas production function in section 2.4.14 ˆ added a section about imposing monotonicity on the Translog input distance function (section 8.5.6) ˆ several minor corrections and improvements

3

Contents 1 Introduction

13

1.1

Objectives of the course and the lecture notes . . . . . . . . . . . . . . . . . . . . . 13

1.2

An extremely short introduction to R . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.2.1

Some commands for simple calculations . . . . . . . . . . . . . . . . . . . . 14

1.2.2

Creating objects and assigning values . . . . . . . . . . . . . . . . . . . . . 15

1.2.3

Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.2.4

Simple functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

1.2.5

Comparing values and Boolean values . . . . . . . . . . . . . . . . . . . . . 18

1.2.6

Data sets (“data frames”) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1.2.7

Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.2.8

Simple graphics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

1.2.9

Other useful commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1.2.10 Extension packages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.2.11 Reading data into R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.2.12 Linear regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.3

R packages

1.4

Data sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.1

1.4.2

1.5

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

French apple producers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.1.1

Description of the data set . . . . . . . . . . . . . . . . . . . . . . 30

1.4.1.2

Abbreviating name of data set . . . . . . . . . . . . . . . . . . . . 31

1.4.1.3

Calculation of input quantities . . . . . . . . . . . . . . . . . . . . 32

1.4.1.4

Calculation of total costs, variable costs, and cost shares . . . . . 32

1.4.1.5

Calculation of profit and gross margin . . . . . . . . . . . . . . . . 33

Rice producers on the Philippines

. . . . . . . . . . . . . . . . . . . . . . . 33

1.4.2.1

Description of the data set . . . . . . . . . . . . . . . . . . . . . . 33

1.4.2.2

Mean-scaling quantities . . . . . . . . . . . . . . . . . . . . . . . . 34

1.4.2.3

Logarithmic mean-scaled quantities . . . . . . . . . . . . . . . . . 34

1.4.2.4

Mean-adjusting the time trend . . . . . . . . . . . . . . . . . . . . 35

1.4.2.5

Total costs and cost shares . . . . . . . . . . . . . . . . . . . . . . 35

1.4.2.6

Specifying panel structure . . . . . . . . . . . . . . . . . . . . . . . 35

Mathematical and statistical methods . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.5.1

Aggregating quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

1.5.2

Concave and convex functions . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4

Contents 1.5.3

Quasiconcave and quasiconvex functions . . . . . . . . . . . . . . . . . . . . 38

1.5.4

Delta method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

2 Primal Approach: Production Function 2.1

40

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.1.1

Production function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.1.2

Average products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.1.3

Total factor productivity

2.1.4

Marginal products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.1.5

Output elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

2.1.6

Elasticity of scale and most productive scale size . . . . . . . . . . . . . . . 41

2.1.7

Marginal rates of technical substitution . . . . . . . . . . . . . . . . . . . . 42

2.1.8

Relative marginal rates of technical substitution . . . . . . . . . . . . . . . 43

2.1.9

Elasticities of substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

2.1.9.1

Direct elasticities of substitution . . . . . . . . . . . . . . . . . . . 43

2.1.9.2

Allen elasticities of substitution . . . . . . . . . . . . . . . . . . . 43

2.1.9.3

Morishima elasticities of substitution . . . . . . . . . . . . . . . . 44

2.1.10 Profit maximization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1.11 Cost minimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.1.12 Derived input demand functions and output supply functions . . . . . . . . 48 2.1.12.1 Derived from profit maximization . . . . . . . . . . . . . . . . . . 48 2.1.12.2 Derived from cost minimization . . . . . . . . . . . . . . . . . . . 49 2.1.13 Suitability of the production function for econometric applications . . . . . 49 2.2

2.3

Productivity measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.2.1

Average products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

2.2.2

Total factor productivity

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

Linear production function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.3.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.3.2

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.3.3

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

2.3.4

Predicted output quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

2.3.5

Marginal products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3.6

Output elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2.3.7

Elasticity of scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.3.8

Marginal rates of technical substitution . . . . . . . . . . . . . . . . . . . . 64

2.3.9

Relative marginal rates of technical substitution . . . . . . . . . . . . . . . 65

2.3.10 First-order conditions for profit maximization . . . . . . . . . . . . . . . . . 65 2.3.11 First-order conditions for cost minimization . . . . . . . . . . . . . . . . . . 68 2.3.12 Derived input demand functions and output supply functions . . . . . . . . 70

5

Contents 2.4

Cobb-Douglas production function . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.4.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.4.2

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.4.3

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.4.4

Predicted output quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

2.4.5

Output elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

2.4.6

Marginal products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

2.4.7

Elasticity of scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

2.4.8

Marginal rates of technical substitution . . . . . . . . . . . . . . . . . . . . 78

2.4.9

Relative marginal rates of technical substitution . . . . . . . . . . . . . . . 79

2.4.10 First and second partial derivatives . . . . . . . . . . . . . . . . . . . . . . . 80 2.4.11 Elasticities of substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 2.4.11.1 Direct elasticities of substitution . . . . . . . . . . . . . . . . . . . 81 2.4.11.2 Allen elasticities of substitution . . . . . . . . . . . . . . . . . . . 82 2.4.11.3 Morishima elasticities of substitution . . . . . . . . . . . . . . . . 86 2.4.12 Quasiconcavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.4.13 First-order conditions for profit maximization . . . . . . . . . . . . . . . . . 88 2.4.14 First-order conditions for cost minimization . . . . . . . . . . . . . . . . . . 90 2.4.15 Derived input demand functions and output supply functions . . . . . . . . 93 2.4.16 Derived input demand elasticities . . . . . . . . . . . . . . . . . . . . . . . . 96 2.5

Quadratic production function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.5.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

2.5.2

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

2.5.3

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

2.5.4

Predicted output quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

2.5.5

Marginal products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

2.5.6

Output elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

2.5.7

Elasticity of scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

2.5.8

Marginal rates of technical substitution . . . . . . . . . . . . . . . . . . . . 105

2.5.9

Relative marginal rates of technical substitution . . . . . . . . . . . . . . . 107

2.5.10 Quasiconcavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2.5.11 Elasticities of substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 2.5.11.1 Direct elasticities of substitution . . . . . . . . . . . . . . . . . . . 111 2.5.11.2 Allen elasticities of substitution . . . . . . . . . . . . . . . . . . . 113 2.5.11.3 Comparison of direct and Allen elasticities of substitution . . . . . 115 2.5.12 First-order conditions for profit maximization . . . . . . . . . . . . . . . . . 116 2.5.13 First-order conditions for cost minimization . . . . . . . . . . . . . . . . . . 116 2.6

Translog production function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 2.6.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6

Contents 2.6.2

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

2.6.3

Statistical significance of individual inputs . . . . . . . . . . . . . . . . . . . 122

2.6.4

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

2.6.5

Predicted output quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

2.6.6

Output elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

2.6.7

Marginal products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

2.6.8

Elasticity of scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

2.6.9

Marginal rates of technical substitution . . . . . . . . . . . . . . . . . . . . 131

2.6.10 Relative marginal rates of technical substitution . . . . . . . . . . . . . . . 133 2.6.11 Second partial derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 2.6.12 Quasiconcavity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 2.6.13 Elasticities of substitution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138 2.6.13.1 Direct elasticities of substitution . . . . . . . . . . . . . . . . . . . 138 2.6.13.2 Allen elasticities of substitution . . . . . . . . . . . . . . . . . . . 141 2.6.13.3 Comparison of direct and Allen elasticities of substitution . . . . . 143 2.6.14 Mean-scaled quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 2.6.15 First-order conditions for profit maximization . . . . . . . . . . . . . . . . . 147 2.6.16 First-order conditions for cost minimization . . . . . . . . . . . . . . . . . . 149 2.7

2.8

Evaluation of different functional forms . . . . . . . . . . . . . . . . . . . . . . . . 152 2.7.1

Goodness of fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152

2.7.2

Test for functional form misspecification . . . . . . . . . . . . . . . . . . . . 154

2.7.3

Theoretical consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155

2.7.4

Plausible estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

2.7.5

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

Non-parametric production function . . . . . . . . . . . . . . . . . . . . . . . . . . 157

3 Dual Approach: Cost Functions 3.1

3.2

164

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 3.1.1

Cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

3.1.2

Properties of the cost function . . . . . . . . . . . . . . . . . . . . . . . . . 164

3.1.3

Cost flexibility and elasticity of size . . . . . . . . . . . . . . . . . . . . . . 164

3.1.4

Homotheticity of cost functions . . . . . . . . . . . . . . . . . . . . . . . . . 165

3.1.5

Short-run cost functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

3.1.6

Cost functions with multiple outputs . . . . . . . . . . . . . . . . . . . . . . 166

3.1.7

Suitability of the cost function for econometric applications . . . . . . . . . 167

Cobb-Douglas cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168 3.2.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

3.2.2

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

3.2.3

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169

3.2.4

Estimation with linear homogeneity in input prices imposed . . . . . . . . . 170

7

Contents 3.2.5

Checking concavity in input prices . . . . . . . . . . . . . . . . . . . . . . . 173

3.2.6

Homotheticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

3.2.7

Optimal input quantities

3.2.8

Optimal cost shares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

3.2.9

Derived input demand functions . . . . . . . . . . . . . . . . . . . . . . . . 181

. . . . . . . . . . . . . . . . . . . . . . . . . . . . 179

3.2.10 Derived input demand elasticities . . . . . . . . . . . . . . . . . . . . . . . . 182 3.2.11 Cost flexibility and elasticity of size . . . . . . . . . . . . . . . . . . . . . . 185 3.2.12 Marginal costs, average costs, and total costs . . . . . . . . . . . . . . . . . 185 3.3

3.4

3.5

Cobb-Douglas short-run cost function . . . . . . . . . . . . . . . . . . . . . . . . . 189 3.3.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

3.3.2

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189

3.3.3

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190

3.3.4

Estimation with linear homogeneity in input prices imposed . . . . . . . . . 190

Cobb-Douglas cost function with multiple outputs . . . . . . . . . . . . . . . . . . 192 3.4.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

3.4.2

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

3.4.3

Cost flexibility and elasticity of size . . . . . . . . . . . . . . . . . . . . . . 192

Translog cost function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192 3.5.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192

3.5.2

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193

3.5.3

Linear homogeneity in input prices . . . . . . . . . . . . . . . . . . . . . . . 194

3.5.4

Estimation with linear homogeneity in input prices imposed . . . . . . . . . 196

3.5.5

Cost flexibility and elasticity of size . . . . . . . . . . . . . . . . . . . . . . 200

3.5.6

Homotheticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202

3.5.7

Marginal costs and average costs . . . . . . . . . . . . . . . . . . . . . . . . 204

3.5.8

Derived input demand functions . . . . . . . . . . . . . . . . . . . . . . . . 208

3.5.9

Derived input demand elasticities . . . . . . . . . . . . . . . . . . . . . . . . 210

3.5.10 Theoretical consistency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 3.6

Translog cost function with multiple outputs . . . . . . . . . . . . . . . . . . . . . 216 3.6.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

3.6.2

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

4 Dual Approach: Profit Function 4.1

218

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218 4.1.1

Profit functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

4.1.2

Short-run profit functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218

4.2

Graphical illustration of profit and gross margin . . . . . . . . . . . . . . . . . . . 218

4.3

Cobb-Douglas profit function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 4.3.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220

4.3.2

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

8

Contents

4.4

4.3.3

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221

4.3.4

Estimation with linear homogeneity in all prices imposed . . . . . . . . . . 222

4.3.5

Checking convexity in all prices . . . . . . . . . . . . . . . . . . . . . . . . . 224

4.3.6

Predicted profit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230

4.3.7

Optimal profit shares

4.3.8

Derived output supply and input demand functions

4.3.9

Derived output supply and input demand elasticities . . . . . . . . . . . . . 233

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231

Cobb-Douglas short-run profit function

5.2

. . . . . . . . . . . . . . . . . . . . . . . . 235

4.4.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

4.4.2

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235

4.4.3

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236

4.4.4

Estimation with linear homogeneity in all prices imposed . . . . . . . . . . 236

4.4.5

Returns to scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

4.4.6

Shadow prices of quasi-fixed inputs . . . . . . . . . . . . . . . . . . . . . . . 238

5 Efficiency Measures 5.1

. . . . . . . . . . . . . 233

240

Technical efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 5.1.1

Output-oriented technical efficiency with one output . . . . . . . . . . . . . 240

5.1.2

Input-oriented technical efficiency with one input . . . . . . . . . . . . . . . 240

5.1.3

Output-oriented technical efficiency with two or more outputs . . . . . . . . 240

5.1.4

Input-oriented technical efficiency with two or more inputs . . . . . . . . . 241

Allocative efficiency, revenue efficiency, cost efficiency . . . . . . . . . . . . . . . . . 241 5.2.1

Output-oriented allocative efficiency and revenue efficiency . . . . . . . . . 241

5.2.2

Input-oriented allocative efficiency and cost efficiency . . . . . . . . . . . . 242

5.3

Profit efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

5.4

Scale efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243

6 Stochastic Frontier Analysis 6.1

244

Stochastic production frontiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 6.1.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244 6.1.1.1

Marginal products and output elasticities in SFA models . . . . . 246

6.1.2

Skewness of residuals from OLS estimations . . . . . . . . . . . . . . . . . . 246

6.1.3

Cobb-Douglas stochastic production frontier . . . . . . . . . . . . . . . . . . 247

6.1.4

6.1.3.1

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247

6.1.3.2

Variance parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 248

6.1.3.3

Statistical tests for inefficiencies . . . . . . . . . . . . . . . . . . . 250

6.1.3.4

Obtaining technical efficiency estimates . . . . . . . . . . . . . . . 251

6.1.3.5

Truncated normal distribution of the inefficiency term . . . . . . . 253

Translog production frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . 255 6.1.4.1

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255

9

Contents

6.1.5 6.2

6.3

6.4

6.1.4.2

Statistical test for inefficiencies . . . . . . . . . . . . . . . . . . . . 256

6.1.4.3

Testing against the Cobb-Douglas functional form . . . . . . . . . 256

6.1.4.4

Obtaining technical efficiency estimates . . . . . . . . . . . . . . . 257

6.1.4.5

Truncated normal distribution of the inefficiency term . . . . . . . 258

Translog production frontier with mean-scaled variables . . . . . . . . . . . 261

Stochastic cost frontiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262 6.2.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262

6.2.2

Skewness of residuals from OLS estimations . . . . . . . . . . . . . . . . . . 263

6.2.3

Estimation of a Cobb-Douglas stochastic cost frontier . . . . . . . . . . . . 264

6.2.4

Short-run cost frontiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266

6.2.5

Profit frontiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 268

Analyzing the effects of z variables . . . . . . . . . . . . . . . . . . . . . . . . . . . 268 6.3.1

Production functions with z variables . . . . . . . . . . . . . . . . . . . . . 269

6.3.2

Production frontiers with z variables . . . . . . . . . . . . . . . . . . . . . . 270

6.3.3

Efficiency effects production frontiers . . . . . . . . . . . . . . . . . . . . . . 273

Decomposition of cost efficiency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277

7 Data Envelopment Analysis (DEA)

281

7.1

Preparations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281

7.2

DEA with input-oriented efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . . 281

7.3

DEA with output-oriented efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . 284

7.4

DEA with “super efficiencies” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284

7.5

DEA with graph hyperbolic efficiencies . . . . . . . . . . . . . . . . . . . . . . . . . 284

8 Distance Functions 8.1

Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 8.1.1

8.1.2

8.2

285

Output distance functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285 8.1.1.1

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285

8.1.1.2

Distance elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . 291

8.1.1.3

Elasticity of scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 292

8.1.1.4

Optimising behaviour and distance elasticities . . . . . . . . . . . 292

8.1.1.5

Obtaining unobserved output prices . . . . . . . . . . . . . . . . . 294

Input distance functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294 8.1.2.1

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 295

8.1.2.2

Distance elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . 301

8.1.2.3

Elasticity of scale . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

Cobb-Douglas output distance function . . . . . . . . . . . . . . . . . . . . . . . . 301 8.2.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301

8.2.2

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 302

8.2.3

Distance elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

10

Contents 8.2.4

Elasticity of scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304

8.2.5

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305

8.2.6 8.3

8.2.5.2

Non-decreasing in output quantities . . . . . . . . . . . . . . . . . 305

8.2.5.3

Quasiconvex in input quantities . . . . . . . . . . . . . . . . . . . 305

8.2.5.4

Convex in output quantities . . . . . . . . . . . . . . . . . . . . . 308

Efficiency estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310

8.3.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

8.3.2

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

8.3.3

Distance elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316

8.3.4

Elasticity of scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317

8.3.5

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319 8.3.5.1

Non-increasing in input quantities . . . . . . . . . . . . . . . . . . 319

8.3.5.2

Non-decreasing in output quantities . . . . . . . . . . . . . . . . . 319

8.3.5.3

Quasiconvex in input quantities . . . . . . . . . . . . . . . . . . . 320

8.3.5.4

Convex in output quantities . . . . . . . . . . . . . . . . . . . . . 323

Efficiency estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326

Cobb-Douglas input distance function . . . . . . . . . . . . . . . . . . . . . . . . . 327 8.4.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

8.4.2

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327

8.4.3

Distance elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329

8.4.4

Elasticity of scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

8.4.5

Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331

8.4.6 8.5

Non-increasing in input quantities . . . . . . . . . . . . . . . . . . 305

Translog output distance function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 311

8.3.6 8.4

8.2.5.1

8.4.5.1

Non-decreasing in input quantities . . . . . . . . . . . . . . . . . . 331

8.4.5.2

Non-increasing in output quantities . . . . . . . . . . . . . . . . . 331

8.4.5.3

Concave in input quantities . . . . . . . . . . . . . . . . . . . . . . 331

8.4.5.4

Quasiconcave in output quantities . . . . . . . . . . . . . . . . . . 334

Efficiency estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336

Translog input distance function . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 8.5.1

Specification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

8.5.2

Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337

8.5.3

Distance elasticities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342

8.5.4

Elasticity of scale . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343

8.5.5

Efficiency estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

8.5.6

Imposing monotonicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344

11

Contents 9 Panel Data and Technological Change 9.1

350

Average production functions with technological change . . . . . . . . . . . . . . . 350 9.1.1

Cobb-Douglas production function with technological change . . . . . . . . 350 9.1.1.1

Pooled estimation of the Cobb-Douglas production function with technological change . . . . . . . . . . . . . . . . . . . . . . . . . . 351

9.1.1.2

Panel data estimations of the Cobb-Douglas production function with technological change . . . . . . . . . . . . . . . . . . . . . . . 352

9.1.2

Translog production function with constant and neutral technological change357 9.1.2.1

Pooled estimation of the Translog production function with constant and neutral technological change . . . . . . . . . . . . . . . . 357

9.1.2.2

Panel-data estimations of the Translog production function with constant and neutral technological change . . . . . . . . . . . . . . 359

9.1.3

Translog production function with non-constant and non-neutral technological change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364 9.1.3.1

Pooled estimation of a translog production function with nonconstant and non-neutral technological change . . . . . . . . . . . 365

9.1.3.2

Panel-data estimations of a Translog production function with non-constant and non-neutral technological change . . . . . . . . . 370

9.2

Frontier production functions with technological change . . . . . . . . . . . . . . . 376 9.2.1

9.2.2

Cobb-Douglas production frontier with technological change . . . . . . . . . 376 9.2.1.1

Time-invariant individual efficiencies . . . . . . . . . . . . . . . . . 377

9.2.1.2

Time-variant individual efficiencies . . . . . . . . . . . . . . . . . . 380

9.2.1.3

Observation-specific efficiencies . . . . . . . . . . . . . . . . . . . . 384

9.2.1.4

Fixed effects in the production frontier . . . . . . . . . . . . . . . 386

Translog production frontier with constant and neutral technological change 389 9.2.2.1

9.2.3

Observation-specific efficiencies . . . . . . . . . . . . . . . . . . . . 389

Translog production frontier with non-constant and non-neutral technological change . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392 9.2.3.1

9.2.4

Observation-specific efficiencies . . . . . . . . . . . . . . . . . . . . 392

Decomposition of productivity growth . . . . . . . . . . . . . . . . . . . . . 395

9.3

Analyzing productivity growths with Data Envelopment Analysis (DEA) . . . . . . 396

9.4

Cost functions with technical change . . . . . . . . . . . . . . . . . . . . . . . . . . 397 9.4.1

Translog cost function with constant and neutral technical change . . . . . 397

9.4.2

Translog cost function with non-constant and non-neutral technical change 397

12

1 Introduction 1.1 Objectives of the course and the lecture notes Knowledge about production technologies and producer behavior is important for politicians, business organizations, government administrations, financial institutions, the EU, and other national and international organizations who desire to know how contemplated policies and market conditions can affect production, prices, income, and resource utilization in agriculture as well as in other industries. The same knowledge is relevant in consultancy of single firms who also want to compare themselves with other firms and their technology with the best practice technology. The participants of my courses in the field of econometric production analysis will obtain relevant theoretical knowledge and practical skills so that they can contribute to the knowledge about production technologies and producer behavior. After completing my courses in the field of econometric production analysis, the students should be able to: ˆ use econometric production analysis and efficiency analysis to analyze various real-world

questions, ˆ interpret the results of econometric production analyses and efficiency analyses, ˆ choose a relevant approach for econometric production and efficiency analysis, and ˆ critically evaluate the appropriateness of a specific econometric production analysis or efficiency analysis for analyzing a specific real-world question. These lecture notes focus on practical applications of econometrics and microeconomic production theory. Hence, they complement textbooks in microeconomic production theory (rather than substituting them).

1.2 An extremely short introduction to R Many tutorials for learning R are freely available on-line, e.g. the official “Introduction to R” (http://cran.r-project.org/doc/manuals/r-release/R-intro.pdf) or the many tutorials listed in the category“Contributed Documentation”(http://cran.r-project.org/other-docs. html). Furthermore, many good books are available, e.g. “A Beginner’s Guide to R” (Zuur, Ieno, and Meesters, 2009), “R Cookbook” (Teetor, 2011), or “Applied Econometrics with R” (Kleiber and Zeileis, 2008).

13

1 Introduction

1.2.1 Some commands for simple calculations R is my favorite “pocket calculator”. . . > 2 + 3 [1] 5 > 2 - 3 [1] -1 > 2 * 3 [1] 6 > 2 / 3 [1] 0.6666667 > 2^3 [1] 8 R uses the standard order of evaluation (as in mathematics). One can use parenthesis (round brackets) to change the order of evaluation. > 2 + 3 * 4^2 [1] 50 > 2 + ( 3 * ( 4^2 ) ) [1] 50 > ( ( 2 + 3 ) * 4 )^2 [1] 400 In R, the hash symbol (#) can be used to add comments to the code, because the hash symbol and all following characters in the same line are ignored by R. > sqrt(2)

# square root

[1] 1.414214 > 2^(1/2)

# the same

14

1 Introduction [1] 1.414214 > 2^0.5

# also the same

[1] 1.414214 > log(3)

# natural logarithm

[1] 1.098612 > exp(3)

# exponential function

[1] 20.08554 The commands can span multiple lines. They are executed as soon as the command can be considered as complete. > 2 + +

3

[1] 5 > ( 2 +

+

+

3 )

[1] 5

1.2.2 Creating objects and assigning values > a a [1] 2 > b b [1] 3 > a * b [1] 6 Initially, the arrow symbol ( a = 4 > a [1] 4 > b = 5 > b [1] 5 > a * b [1] 20 In these lecture notes, I stick to the traditional assignment operator, i.e. the arrow symbol ( A

# NOT the same as "a"

> B

# NOT the same as "b"

> Log(3)

# NOT the same as "log(3)"

> LOG(3)

# NOT the same as "log(3)"

1.2.3 Vectors > v v [1] 1 2 3 4 > 2 + v

# adding 2 to each element

[1] 3 4 5 6 > 2 * v

# multiplying each element by 2

[1] 2 4 6 8 > log( v )

# the natural logarithm of each element

[1] 0.0000000 0.6931472 1.0986123 1.3862944 > w w

16

1 Introduction [1]

2

4

> v + w [1]

3

# element-wise addition 6 11 20

> v * w [1]

8 16

2

# element-wise multiplication 8 24 64

> v %*% w

# scalar product (inner product)

[,1] [1,]

98

> w[2]

# select the second element

[1] 4 > w[c(1,3)]

# select the first and the third element

[1] 2 8 > w[2:4] [1]

4

> w[-2] [1]

2

# select the second, third, and fourth element

8 16 # select all but the second element 8 16

> length( w ) [1] 4

1.2.4 Simple functions > sum( w ) [1] 30 > mean( w ) [1] 7.5 > median( w )

17

1 Introduction [1] 6 > min( w ) [1] 2 > max( w ) [1] 16 > which.min( w ) [1] 1 > which.max( w ) [1] 4

1.2.5 Comparing values and Boolean values > a == 2 [1] FALSE > a != 2 [1] TRUE > a > 4 [1] FALSE > a >= 4 [1] TRUE > w > 3 [1] FALSE

TRUE

TRUE

TRUE

> w == 2^(1:4) [1] TRUE TRUE TRUE TRUE > all.equal( w, 2^(1:4) ) [1] TRUE > w > 3 & w < 6 [1] FALSE

# ampersand = and

TRUE FALSE FALSE

> w < 3 | w > 6

# vertical line = or

[1]

TRUE

TRUE FALSE

TRUE

18

1 Introduction

1.2.6 Data sets (“data frames”) The data set “women” is included in R. > data( "women" )

# load the data set into the workspace

> women height weight 1

58

115

2

59

117

3

60

120

4

61

123

5

62

126

6

63

129

7

64

132

8

65

135

9

66

139

10

67

142

11

68

146

12

69

150

13

70

154

14

71

159

15

72

164

> names( women )

# display the variable names

[1] "height" "weight" > dim( women ) [1] 15

# dimension of the data set (rows and columns)

2

> nrow( women )

# number of rows (observations)

[1] 15 > ncol( women )

# number of columns (variables)

[1] 2 > women[[ "height" ]]

# display the values of variable "height"

[1] 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 > women$height

# short-cut for the previous command

19

1 Introduction [1] 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 > women$height[ 3 ]

# height of the third observation

[1] 60 > women[ 3, "height" ]

# the same

[1] 60 > women[ 3, 1 ]

# also the same

[1] 60 > women[ 1:3, 1 ]

# height of the first three observations

[1] 58 59 60 > women[ 1:3, ]

# all variables of the first three observations

height weight 1

58

115

2

59

117

3

60

120

> women$cmHeight women$kgWeight women$bmi women height weight cmHeight kgWeight

bmi

1

58

115

147.32 52.15420 24.03067

2

59

117

149.86 53.06122 23.62685

3

60

120

152.40 54.42177 23.43164

4

61

123

154.94 55.78231 23.23643

5

62

126

157.48 57.14286 23.04152

6

63

129

160.02 58.50340 22.84718

7

64

132

162.56 59.86395 22.65364

8

65

135

165.10 61.22449 22.46110

9

66

139

167.64 63.03855 22.43112

10

67

142

170.18 64.39909 22.23631

11

68

146

172.72 66.21315 22.19520

12

69

150

175.26 68.02721 22.14711

13

70

154

177.80 69.84127 22.09269

14

71

159

180.34 72.10884 22.17198

15

72

164

182.88 74.37642 22.23836

20

# new variable: BMI

1 Introduction

1.2.7 Functions In order to execute a function in R, the function name has to be followed by a pair of parenthesis (round brackets). The documentation of a function (if available) can be obtained by, e.g., typing at the R prompt a question mark followed by the name of the function. > ?log One can read in the documentation of the function log, e.g., that this function has a second optional argument base, which can be used to specify the base of the logarithm. By default, the base is equal to the Euler number (e, exp(1)). A different base can be chosen by adding a second argument, either with or without specifying the name of the argument. > log( 100, base = 10 ) [1] 2 > log( 100, 10 ) [1] 2

1.2.8 Simple graphics Histograms can be created with the command hist. The optional argument breaks can be used to specify the approximate number of cells: > hist( women$bmi )

4 3 0

1

2

Frequency

4 0

2

Frequency

6

8

> hist( women$bmi, breaks = 10 )

22.0

22.5

23.0

23.5

24.0

24.5

22.0

women$bmi

22.5

23.0

23.5

women$bmi

Figure 1.1: Histogram of BMIs The resulting histogram is shown in figure 1.1. Scatter plots can be created with the command plot: > plot( women$height, women$weight ) The resulting scatter plot is shown in figure 1.2.

21

24.0

120

130

140

women$weight

150

160

1 Introduction

58

60

62

64

66

68

70

72

women$height

Figure 1.2: Scatter plot of heights and weights

1.2.9 Other useful commands > class( a ) [1] "numeric" > class( women ) [1] "data.frame" > class( women$height ) [1] "numeric" > ls() [1] "a" > rm(w)

# list all objects in the workspace "b"

"v"

"w"

"women"

# remove an object

> ls() [1] "a"

"b"

"v"

"women"

1.2.10 Extension packages Currently (June 12, 2013, 2 pm GMT), 4611 extension packages for R are available on CRAN (Comprehensive R Archive Network, http://cran.r-project.org). When an extension package is installed, it can be loaded with the command library. The following command loads the R package foreign that includes function for reading data in various formats. > library( "foreign" )

22

1 Introduction Please note that you should cite scientific software packages in your publications if you used them for obtaining your results (as any other scientific works). You can use the command citation to find out how an R package should be cited, e.g.: > citation( "frontier" ) To cite package 'frontier' in publications use: Tim Coelli and Arne Henningsen (2020). frontier: Stochastic Frontier Analysis. R package version 1.1-9. https://CRAN.R-Project.org/package=frontier. A BibTeX entry for LaTeX users is @Manual{, title = {frontier: Stochastic Frontier Analysis}, author = {Tim Coelli and Arne Henningsen}, year = {2020}, note = {R package version 1.1-9}, url = {https://CRAN.R-Project.org/package=frontier}, }

1.2.11 Reading data into R R can read and import data from many different file formats. This is described in the official R manual “R Data Import/Export” (http://cran.r-project.org/doc/manuals/r-release/ R-data.pdf). I usually read my data into R from files in CSV (comma separated values) format. This can be done by the function read.csv. The command read.csv2 can read files in the “European CSV format” (values separated by semicolons, comma as decimal separator). The functions read.spss and read.xport (both in package foreign) can read SPSS data files and SAS “XPORT” files, respectively. While the add-on package readstata13 can be used to read binary data files from all STATA versions (including versions 13 and 14), function read.dta (in package foreign) can only read binary data files from STATA versions 5–12. Functions for reading MS-Excel files are available, e.g., in the packages XLConnect and xlsx.

1.2.12 Linear regression The command for estimating linear models in R is lm. The first argument of the command lm specifies the model that should be estimated. This must be a formula object that consists of the name of the dependent variable, followed by a tilde (~) and the name of the explanatory variable. Argument data can be used to specify the data set:

23

1 Introduction > olsWeight olsWeight Call: lm(formula = weight ~ height, data = women) Coefficients: (Intercept)

height

-87.52

3.45

The summary method can be used to display summary statistics of the regression: > summary( olsWeight ) Call: lm(formula = weight ~ height, data = women) Residuals: Min

1Q

Median

3Q

Max

-1.7333 -1.1333 -0.3833

0.7417

3.1167

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -87.51667

5.93694

-14.74 1.71e-09 ***

height

0.09114

37.85 1.09e-14 ***

3.45000

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1.525 on 13 degrees of freedom Multiple R-squared: F-statistic:

0.991,

Adjusted R-squared:

1433 on 1 and 13 DF,

p-value: 1.091e-14

0.9903

The command abline can be used to add a linear (regression) line to a (scatter) plot: > plot( women$height, women$weight ) > abline( olsWeight ) The resulting plot is shown in figure 1.3. This figure indicates that the relationship between the height and the corresponding average weights of the women is slightly nonlinear. Therefore, we add the squared height as additional explanatory regressor. When specifying more than one explanatory variable, the names of the explanatory variables must be separated by plus signs (+):

24

120

130

140

women$weight

150

160

1 Introduction

58

60

62

64

66

68

70

72

women$height

Figure 1.3: Scatter plot of heights and weights with estimated regression line > women$heightSquared olsWeight2 summary( olsWeight2 ) Call: lm(formula = weight ~ height + heightSquared, data = women) Residuals: Min

1Q

Median

3Q

Max

-0.50941 -0.29611 -0.00941

0.28615

0.59706

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

261.87818

25.19677

10.393 2.36e-07 ***

-7.34832

0.77769

-9.449 6.58e-07 ***

0.08306

0.00598

13.891 9.32e-09 ***

height heightSquared --Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3841 on 12 degrees of freedom Multiple R-squared:

0.9995,

Adjusted R-squared:

F-statistic: 1.139e+04 on 2 and 12 DF,

0.9994

p-value: < 2.2e-16

One can use the function I() to calculate explanatory variables directly in the formula: > olsWeight3 summary( olsWeight3 )

25

1 Introduction Call: lm(formula = weight ~ height + I(height^2), data = women) Residuals: Min

1Q

Median

3Q

Max

-0.50941 -0.29611 -0.00941

0.28615

0.59706

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) 261.87818 height

25.19677

10.393 2.36e-07 ***

-7.34832

0.77769

-9.449 6.58e-07 ***

0.08306

0.00598

13.891 9.32e-09 ***

I(height^2) ---

Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3841 on 12 degrees of freedom Multiple R-squared:

0.9995,

Adjusted R-squared:

F-statistic: 1.139e+04 on 2 and 12 DF,

0.9994

p-value: < 2.2e-16

The coef method for lm objects can be used to extract the vector of the estimated coefficients: > coef( olsWeight2 ) (Intercept) 261.87818358

height heightSquared -7.34831933

0.08306399

When the coef method is applied to the object returned by the summary method for lm objects, the matrix of the estimated coefficients, their standard errors, their t-values, and their P -values is returned: > coef( summary( olsWeight2 ) ) Estimate (Intercept) height heightSquared

Std. Error

t value

Pr(>|t|)

261.87818358 25.196770820 10.393323 2.356879e-07 -7.34831933

0.777692280 -9.448878 6.584476e-07

0.08306399

0.005979642 13.891131 9.322439e-09

The variance covariance matrix of the estimated coefficients can be obtained by the vcov method: > vcov( olsWeight2 )

26

1 Introduction (Intercept)

height heightSquared

(Intercept)

634.8772597 -19.586524729

height

-19.5865247

heightSquared

0.1504022

1.504022e-01

0.604805283 -4.648296e-03 -0.004648296

3.575612e-05

The residuals method for lm objects can be used to obtain the residuals: > residuals( olsWeight2 ) 1

2

3

4

5

6

-0.102941176 -0.473109244 -0.009405301

0.288170653

0.419618617

0.384938591

10

11

12

7

8

0.184130575 -0.182805430

9

0.284130575 -0.415061409 -0.280381383 -0.311829347

13

14

15

-0.509405301

0.126890756

0.597058824

The fitted method for lm objects can be used to obtain the fitted values: > fitted( olsWeight2 ) 1

2

3

4

5

6

7

8

115.1029 117.4731 120.0094 122.7118 125.5804 128.6151 131.8159 135.1828 9

10

11

12

13

14

15

138.7159 142.4151 146.2804 150.3118 154.5094 158.8731 163.4029 We can evaluate the “fit” of the model by plotting the fitted values against the observed values of the dependent variable and adding a 45-degree line: > plot( women$weight, fitted( olsWeight2 ) ) > abline(0,1) The resulting scatter plot is shown in figure 1.4. The plot method for lm objects can be used to generate diagnostic plots > plot( olsWeight2 ) The resulting diagnostic plots are shown in figure 1.5.

1.3 R packages The following R packages are particularly relevant for empirical production, efficiency, and productivity analysis using R (in alphabetical order):

27

150 140 130 120

fitted(olsWeight2)

160

1 Introduction

120

130

140

150

160

women$weight

Figure 1.4: Observed and fitted values of the dependent variable

2

120

13

130

140

150

0

1

2

15

−1

0.2

15

Standardized residuals

Normal Q−Q

−0.2 −0.6

Residuals

0.6

Residuals vs Fitted

13

160

−1

Fitted values

Scale−Location

1.5

0

1

Theoretical Quantiles

Residuals vs Leverage 2 1

1.0 0.5

15

0

13

2

−1

Standardized residuals

15

13 Cook's distance

0.0

Standardized residuals

2

120

130

140

150

160

0.0

Fitted values

0.1

0.2

2

0.3

Leverage

Figure 1.5: Diagnostic plots

28

0.4

1 Introduction Benchmarking ˆ ˆ ˆ ˆ

various DEA models rudimentary SFA models extensively described in the book of Bogetoft and Otto (2011) available at http://cran.r-project.org/package=Benchmarking

FEAR ˆ ˆ ˆ ˆ ˆ

various DEA models bootstrapping closed source non-academic use requires payment of a license fee available at: http://www.clemson.edu/economics/faculty/wilson/Software/FEAR/fear. html

frontiles ˆ alpha-quantile and order-m efficiency ˆ available at http://cran.r-project.org/package=frontiles

frontier ˆ cross-sectional and panel data stochastic frontier models as suggested by Battese and Coelli

(1992) ˆ stochastic frontier models incorporating a model for inefficiency effects as suggested by Battese and Coelli (1995) ˆ based on FRONTIER 4.1 (Coelli, 1996) ˆ available at http://cran.r-project.org/package=frontier nonparaeff ˆ DEA ˆ available at http://cran.r-project.org/package=nonparaeff

np ˆ nonparametric kernel regression ˆ available at http://cran.r-project.org/package=np

plm ˆ estimation and testing of various specifications of linear panel data models ˆ available at http://cran.r-project.org/package=plm

29

1 Introduction rDEA ˆ robust DEA ˆ available at http://cran.r-project.org/package=rDEA

semsfa ˆ semiparametric SFA ˆ available at http://cran.r-project.org/package=semsfa

sfa ˆ SFA ˆ available at http://cran.r-project.org/package=sfa

spfrontier ˆ spatial SFA ˆ available at http://cran.r-project.org/package=spfrontier

ssfa ˆ spatial SFA ˆ available at http://cran.r-project.org/package=ssfa

TFDEA ˆ technology forecasting using DEA ˆ available at http://cran.r-project.org/package=TFDEA

1.4 Data sets In my courses in the field of econometric production analysis, I usually use two data sets: one cross-sectional data set of French apple producers and a panel data set of rice producers on the Philippines.

1.4.1 French apple producers 1.4.1.1 Description of the data set In this course, we will predominantly use a cross-sectional production data set of 140 French apple producers from the year 1986. These data are extracted from a panel data set that has been used in an article published by Ivaldi et al. (1996) in the Journal of Applied Econometrics.

30

1 Introduction The full panel data set is available in the journal’s data archive: http://www.econ.queensu. ca/jae/1996-v11.6/ivaldi-ladoux-ossard-simioni/.1 The cross-sectional data set that we will predominantly use in the course is available in the R package micEcon. It has the name appleProdFr86 and can be loaded by the command: > data( "appleProdFr86", package = "micEcon" ) The names of the variables in the data set can be obtained by the command names: > names( appleProdFr86 ) [1] "vCap"

"vLab"

"vMat"

"qApples"

"qOtherOut" "qOut"

[7] "pCap"

"pLab"

"pMat"

"pOut"

"adv"

The data set includes following variables:2 vCap costs of capital (including land) vLab costs of labor (including remuneration of unpaid family labor) vMat costs of intermediate materials (e.g. seedlings, fertilizer, pesticides, fuel) qOut quantity index of all outputs (apples and other outputs) pCap price index of capital goods pLab price index of labor pMat price index of materials pOut price index of the aggregate output∗ adv use of advisory service∗ Please note that variables indicated by

∗

are not in the original data set but are artificially

generated in order to be able to conduct some further analyses with this data set. Variable names starting with v indicate volumes (values), variable names starting with q indicate quantities, and variable names starting with p indicate prices. 1.4.1.2 Abbreviating name of data set In order to avoid too much typing, give the data set a much shorter name (dat) by creating a copy of the data set and removing the original data set: > dat rm( appleProdFr86 )

1

In order to focus on the microeconomic analysis rather than on econometric issues in panel data analysis, we only use a single year from this panel data set. 2 This information is also available in the documentation of this data set, which can be obtained by the command: help( "appleProdFr86", package = "micEcon" ).

31

1 Introduction 1.4.1.3 Calculation of input quantities Our data set does not contain input quantities but prices and costs (volumes) of the inputs. As we will need to know input quantities for many of our analyses, we calculate input quantity indices based on following identity: v i = x i · wi ,

(1.1)

where wi is the price, xi is the quantity and vi is the volume of the ith input. In R, we can calculate the input quantities with the following commands: > dat$qCap dat$qLab dat$qMat dat$cost all.equal( dat$cost, with( dat, pCap * qCap + pLab * qLab + pMat * qMat ) ) [1] TRUE Variable costs are defined as: cv =

X

wi xi ,

(1.3)

i∈N 1

where N 1 is a vector of the indices of the variable inputs. If capital is a quasi-fixed input and labor and materials are variable inputs, the apple producers’ variable costs can be calculated by following command: > dat$vCost dat$sCap dat$sLab dat$sMat dat$profit all.equal( dat$cost, with( dat, pCap * qCap + pLab * qLab + pMat * qMat ) ) [1] TRUE The gross margin (“variable profit”) is defined as: πv = p y −

X

wi xi = p y − cv ,

(1.5)

i∈N 1

where all variables are defined as above. If capital is a quasi-fixed input and labor and materials are variable inputs, the apple producers’ gross margins can be calculated by following command: > dat$vProfit data( "riceProdPhil", package = "frontier" ) The names of the variables in the data set can be obtained by the command names: > names( riceProdPhil ) [1] "YEARDUM"

"FMERCODE" "PROD"

"AREA"

"LABOR"

"NPK"

[7] "OTHER"

"PRICE"

"AREAP"

"LABORP"

"NPKP"

"OTHERP"

"EDYRS"

"HHSIZE"

"NADULT"

"BANRAT"

[13] "AGE"

The following variables are of particular importance for our analysis:

33

1 Introduction PROD output (tonnes of freshly threshed rice) AREA area planted (hectares). LABOR labor used (man-days of family and hired labor) NPK fertilizer used (kg of active ingredients) YEARDUM time period (1 = 1990, . . . , 8 = 1997) In our analysis of the production technology of the rice producers we will use variable PROD as output quantity and variables AREA, LABOR, and NPK as input quantities. 1.4.2.2 Mean-scaling quantities In some model specifications, it is an advantage to use mean-scaled quantities. Therefore, we create new variables with mean-scaled input and output quantities: > riceProdPhil$area

riceProdPhil$labor riceProdPhil$npk

riceProdPhil$prod

colMeans( riceProdPhil[ , c( "prod", "area", "labor", "npk" ) ] ) prod 1

area labor 1

npk

1

1

> log( colMeans( riceProdPhil[ , c( "prod", "area", "labor", "npk" ) ] ) ) prod

area

labor

npk

0.000000e+00 -1.110223e-16

0.000000e+00

0.000000e+00

1.4.2.3 Logarithmic mean-scaled quantities As we use logarithmic input and output quantities in the Cobb-Douglas and Translog specifications, we can reduce our typing work by creating variables with logarithmic (mean-scaled) input and output quantities: > riceProdPhil$lArea

riceProdPhil$lLabor riceProdPhil$lNpk

riceProdPhil$lProd

colMeans( riceProdPhil[ , c( "lProd", "lArea", "lLabor", "lNpk" ) ] ) lProd

lArea

lLabor

lNpk

-0.3263075 -0.2718549 -0.2772354 -0.4078492 1.4.2.4 Mean-adjusting the time trend In some model specifications, it is an advantage to have a time trend variable that is zero at the sample mean. If we subtract the sample mean from our time trend variable, the sample mean of the adjusted time trend is zero: > riceProdPhil$mYear

mean( riceProdPhil$mYear ) [1] 0 1.4.2.5 Total costs and cost shares The following code calculates total costs and the cost shares (ignoring “other inputs”): > riceProdPhil$cost +

riceProdPhil$sArea

riceProdPhil$sLabor riceProdPhil$sNpk

all.equal( with( riceProdPhil, sArea + sLabor + sNpk ), +

rep( 1, nrow( riceProdPhil ) ) )

[1] TRUE 1.4.2.6 Specifying panel structure This data set does not include any information about its panel structure. Hence, R would ignore the panel structure and treat this data set as cross-sectional data collected from 352 different producers. The command pdata.frame of the plm package (Croissant and Millo, 2008) can be used to create data sets that include the information on its panel structure. The following commands creates a new data set of the rice producers from the Philippines that includes information on the panel structure, i.e. variable FMERCODE indicates the individual (farmer), and variable YEARDUM indicated the time period (year):3 > library( "plm" ) > pdat plot( dat$XP, dat$XL, log = "xy" ) The resulting scatter plots are shown in figure 1.6. As a compromise, one can use the Fisher quantity index, which is the geometric mean of the Paasche quantity index and the Laspeyres quantity index: > dat$X library( "micEconIndex" ) > dat$XP2 all.equal( dat$XP, dat$XP2, check.attributes = FALSE ) [1] TRUE

36

1

0.5

1.0

XL 2

XL

3

2.0

4

5

5.0

1 Introduction

1

2

3

4

5

0.5

1.0

XP

2.0

5.0

XP

Figure 1.6: Comparison of Paasche and Laspeyres quantity indices > dat$XL2 all.equal( dat$XL, dat$XL2, check.attributes = FALSE ) [1] TRUE > dat$X2 all.equal( dat$X, dat$X2, check.attributes = FALSE ) [1] TRUE

1.5.2 Concave and convex functions A function f (x) : RN → R is

concave over a convex set D if and only if convex

θf (xu ) + (1 − θ)f (xv )

≤ f (θxu + (1 − θ)xv ) ≥

for all combinations of xu , xv ∈ D and for all 0 ≤ θ ≤ 1, while it is strictly

(1.7) concave over a convex convex

set D if and only if θf (xu ) + (1 − θ)f (xv )

< f (θxu + (1 − θ)xv ) >

(1.8)

for all combinations of xu , xv ∈ D and for all 0 < θ < 1 (Chiang, 1984, p. 342). concave A continuous and twice continuously differentiable function f (x) : RN → R is over a convex negative convex set D if and only if its Hessian matrix is semidefinite at all x ∈ D, while it is positive concave negative strictly over a convex set D if (but not only if) its Hessian matrix is definite at convex positive all x ∈ D (Chiang, 1984, p. 347).

37

1 Introduction A symmetric quadratic N × N matrix H is negative semidefinite if and only if all its ith-order principal minors (not only its leading principal minors) are non-positive for i being odd and nonnegative for i being even for all i ∈ {1, . . . , N }, while it is positive semidefinite if all its principal minors (not only its leading principal minors) are non-negative (Gantmacher, 1959, p. 307–308). An ith-order principal minor of a symmetric N × N matrix H is the determinant of an k × k submatrix of H, whereas N − k rows and the corresponding N − k columns have been deleted. An N × N matrix has

N
k

ith-order principal minors and in total

PN

k=1

N
k

principal minors.

A quadratic N × N matrix H is negative definite if and only if its first leading principal minor is strictly negative and the following leading principle minors alternate in sign, i.e. |B1 | < 0, |B2 | > 0, |B3 | < 0, . . . , (−1)N |BN | > 0, while it is positive definite if and only if all its N leading principal minors are positive, i.e. |Bi | > 0 ∀ i ∈ {1, . . . , }, where |Bi | is the ith leading principal minor of matrix H (Chiang, 1984, p. 324). An ith-order leading principal minor of a symmetric N × N matrix H is the determinant of an k × k submatrix of H, whereas last N − k rows and the last N − k columns have been deleted. An N × N has N leading principal minors. negative A quadratic N × N matrix H is semidefinite if and only if all N eigenvalues of H are positive non-positive and at least one of these eigenvalues is zero (Chiang, 1984, p. 330). non-negative negative definite if and only if all N eigenvalues of H are A quadratic N × N matrix H is positive negative (Chiang, 1984, p. 330). strictly positive

1.5.3 Quasiconcave and quasiconvex functions A function f (x) : RN → R is

quasiconcave over a convex set D if and only if quasiconvex ≥ f (θxu + (1 − θ)xv ) min(f (xu ), f (xv )) ≤

for all combinations of xu , xv ∈ D and for all 0 ≤ θ ≤ 1, while it is strictly

(1.9) quasiconcave over a quasiconvex

convex set D if and only if > f (θxu + (1 − θ)xv ) min(f (xu ), f (xv ))
< > sufficient ≤ ≤ ≤ ≤ condition for quasiconvexity is that |B1 | 0, |B2 | 0, |B3 | 0, . . . , |BN | 0, where |Bi | is the ith < < < < leading principal minor of the bordered Hessian matrix 



0

f1

f2

...

fN

  f1   B=  f2  .  .. 

f11

f12

...

f12 .. .

f22 .. .

... .. .

f1N  

fN

f1N

f2N

. . . fN N



f2N .. .

 ,    

(1.12)

fi denotes the partial derivative of f (x) with respect to xi , fij denotes the second partial derivative of f (x) with respect to xi and xj , |B1 | is the determinant of the upper left 2 × 2 sub-matrix of B, |B2 | is the determinant of the upper left 3 × 3 sub-matrix of B, . . . , and |BN | is the determinant of B (Chiang, 1984, p. 393f; Chambers, 1988, p. 312). While the first inequality of the necessary condition for quasiconcavity and quasiconvexity, i.e. |B1 | ≤ 0, is always fulfilled given that |B1 | = −f12 , the first inequality of the sufficient condition for quasiconcavity and quasiconvexity, i.e. |B1 | < 0, is only fulfilled if f1 6= 0 (Chiang, 1984, p. 394f).

1.5.4 Delta method If we have estimated a parameter vector β and its variance covariance matrix V ar(β) and we calculate a vector of measures (e.g. elasticities) based on the estimated parameters by z = g(β), we can calculate the approximate variance covariance matrix of z by: 

V ar(z) ≈

∂g(β) ∂β

>

V ar(β)

∂g(β) , ∂β

(1.13)

where ∂g(β)/∂β is the Jacobian matrix of z = g(β) with respect to β and the superscript > is the transpose operator.

39

2 Primal Approach: Production Function 2.1 Theory 2.1.1 Production function The production function y = f (x)

(2.1)

indicates the maximum quantity of a single output (y) that can be obtained with a vector of given input quantities (x). It is usually assumed that production functions fulfill some properties (see Chambers, 1988, p. 9).

2.1.2 Average products Very simple measures to compare the (partial) productivities of different firms are the inputs’ average products. The average product of the ith input is defined as: APi =

f (x) y = xi xi

(2.2)

The more output one firm produces per unit of input, the more productive is this firm and the higher is the corresponding average product. If two firms use identical input quantities, the firm with the larger output quantity is more productive (has a higher average product). And if two firms produce the same output quantity, the firm with the smaller input quantity is more productive (has a higher average product). However, if these two firms use different input combinations, one firm could be more productive regarding the average product of one input, while the other firm could be more productive regarding the average product of another input.

2.1.3 Total factor productivity As average products measure just partial productivities, it is often desirable to calculate total factor productivities (TFP): TFP =

y , X

where X is a quantity index of all inputs (see section 1.5.1).

40

(2.3)

2 Primal Approach: Production Function

2.1.4 Marginal products The marginal productivities of the inputs can be measured by their marginal products. The marginal product of the ith input is defined as: M Pi =

∂f (x) ∂xi

(2.4)

2.1.5 Output elasticities The marginal productivities of the inputs can also be measured by their output elasticities. The output elasticity of the ith input is defined as: εi =

∂f (x) xi MP = ∂xi f (x) AP

(2.5)

In contrast to the marginal products, the changes of the input and output quantities are measured in relative terms so that output elasticities are independent of the units of measurement. Output elasticities are sometimes also called partial output elasticities or partial production elasticities.

2.1.6 Elasticity of scale and most productive scale size The returns of scale of the technology can be measured by the elasticity of scale: ε=

X

εi

(2.6)

i

If the technology has increasing returns to scale (ε > 1), total factor productivity increases when all input quantities are proportionally increased, because the relative increase of the output quantity y is larger than the relative increase of the aggregate input quantity X in equation (2.3). If the technology has decreasing returns to scale (ε < 1), total factor productivity decreases when all input quantities are proportionally increased, because the relative increase of the output quantity y is less than the relative increase of the aggregate input quantity X. If the technology has constant returns to scale (ε = 1), total factor productivity remains constant when all input quantities change proportionally, because the relative change of the output quantity y is equal to the relative change of the aggregate input quantity X. If the elasticity of scale (monotonically) decreases with firm size, the firm has the most productive scale size (MPSS) at the point, where the elasticity of scale is one. In order to prove this, we let x0 indicate the vector of a firm’s initial input quantities and s > 0 indicate a scalar factor by which the firm scales its initial input quantities up or down so that it uses input quantities x = s · x0 . When the firm uses the input quantities x = s · x0 , its total factor productivity is T F P = f (x)/X(x), where f (.) is the production function and X(.) indicates a linearly homogeneous function that aggregates the individual input quantities to a quantity index. Linear homogeneity of X(.) implies that, e.g., X(s · x0 ) = s X(x0 ) and ∂X(s · x0 )/∂s = X(x0 ). In the

41

2 Primal Approach: Production Function following, we want to find the value of the scale factor s that gives the input quantities x = s · x0 that maximize total factor productivity. A necessary condition for a maximum of the total factor productivity is that its partial derivative with respect to s is zero: ∂T F P =0 ∂s   f (x) ∂ X(x) =0 ∂s ∂f (x) ∂X(x) X(x) − f (x) ∂s ∂s = 0 2 (X(x)) ∂f (x) ∂X(s · x0 ) X(x) − f (x) =0 ∂s ∂s ! N X ∂f (x) ∂xi X(s · x0 ) − f (x) X(x0 ) = 0 ∂x ∂s i i=1 N X ∂f (x) 0 x i=1

∂xi

i=1

(2.8)

(2.9) (2.10) (2.11)

!

i

s X(x0 ) = f (x) X(x0 )

N X ∂f (x) s · x0i i=1 N X

(2.7)

(2.12)

=1

(2.13)

∂f (x) x =1 ∂xi f (x)

(2.14)

∂xi f (x)

N X

i = 1

(2.15)

=1

(2.16)

i=1

Hence, a necessary condition for a maximum of the total factor productivity at a point x = s · x0 is that the elasticity of scale is one ( = 1). The sufficient condition for a maximum of the total factor productivity is fulfilled if the second derivative of the total factor productivity with respect to the scale factor is negative at the point x = s · x0 with  = 1. This is the case if the elasticity of scale decreases with the scale factor (∂/∂s < 0), i.e., firms that are smaller than the point where  = 1 have increasing returns to scale, while firms that are larger than the point where  = 1 have decreasing returns to scale.

2.1.7 Marginal rates of technical substitution The marginal rate of technical substitution between input i and input j is (Chambers, 1988, p. 29): M RT Si,j

∂y ∂xi M Pj ∂xj = =− =− ∂y ∂xj M Pi ∂xi

42

(2.17)

2 Primal Approach: Production Function

2.1.8 Relative marginal rates of technical substitution The relative marginal rate of technical substitution between input i and input j is:

RM RT Si,j

∂y ∂xi xj ∂xj =− = ∂y ∂xj xi ∂xi

xj εj y xi = − ε i y

(2.18)

2.1.9 Elasticities of substitution The elasticity of substitution measures the substitutability between two inputs. It is defined as:

σij =





xi M Pj xj  MxPi  i M Pj d M Pi xj

d

d =



xi xj



−M RT Sij xi xj

d (−M RT Sij )

d =



xi xj



d M RT Sij

M RT Sij xi xj

(2.19)

Thus, if input i is substituted for input j so that the input ratio xi /xj increases by σij %, the marginal rate of technical substitution between input i and input j will increase by 1%. 2.1.9.1 Direct elasticities of substitution The direct elasticity of substitution measures the substitutability between two inputs when the quantities of all other inputs are held constant. Hence, the direct elasticities of substitution can be considered as short-run elasticities of substitution (Chambers, 1988, p. 33). It can be calculated by: D σij =

−fi fj (xi fi + xj fj ) 

xi xj fii fj2 − 2fij fi fj + fjj fi2

,

(2.20)

where fi ≡ ∂f (x)/∂xi is the partial derivative of the production function f (x) with respect to the ith input quantity (xi ), and fij ≡ ∂ 2 f (x)/∂xi ∂xj is the second partial derivative of the production function f (x) with respect to the ith and jth input quantity (xi , xj ) (Chambers, 1988, p. 31). As fij = fji (Young’s theorem), the direct elasticity of substitution is symmetric D = σ D ). (σij ji

2.1.9.2 Allen elasticities of substitution The Allen elasticity of substitution is another measure of the substitutability between two inputs. In contrast to the direct elasticity of substitution, the Allen elasticity of substitution takes into account that the quantities of the other inputs can be adjusted as well. Hence, in comparison to the direct elasticities of substitution, the Allen elasticities of substitution can be considered as long-run elasticities of substitution. It can be calculated by: P

σij =

xk Fij , xi xj F k fk

43

(2.21)

2 Primal Approach: Production Function where F is the determinant of the bordered Hessian matrix B with 



0

f1

f2

...

fN

  f1   B=  f2  .  .. 

f11

f12

...

f12 .. .

f22 .. .

... .. .

f1N  

fN

f1N

f2N

. . . fN N



f2N .. .

 ,    

(2.22)

and Fij is the co-factor of fij , i.e.1 Fij = (−1)i+j ·

0

f1

f2

...

fj−1

fj+1

f1

f11

f12

...

f1,j−1

f1,j+1

f2 .. .

f12 .. .

f22 .. .

... .. .

f2,j−1 .. .

f2,j+1 .. .

fi−1 f1,i−1 f2,i−1 . . . fi−1,j−1 fi−1,j+1 fi+1 f1,i+1 f2,i+1 . . . fi+1,j−1 fi+1,j+1 .. .. .. .. .. .. . . . . . . fN

f1N

f2N

...

fj−1,N

fj+1,N

. . . f1N . . . f2N .. .. . . . . . . fi−1,N . . . fi+1,N .. .. . . . . . fN N

...

fN

(2.23)

As the bordered Hessian matrix is symmetric, the co-factors are also symmetric (Fij = Fji ) so that also the Allen elasticities of substitution are symmetric (σij = σji ). In case of two inputs (x = (x1 , x2 )), F =

2f12 f1 f2 − f11 f22

− f22 f12

P

k fk

xk = f1 x1 + f2 x2 , xi xj = x1 x2 , F12 = f1 f2 , and

so that the direct elasticity of substitution is identical to the Allen

D ). elasticity of substitution (σij = σij

Following condition holds for Allen elasticities of substitution: X i

fi xi Ki σij = 0 with Ki = P k fk xk

(2.24)

(see Chambers, 1988, p. 35). 2.1.9.3 Morishima elasticities of substitution The Morishima elasticity of substitution is a third measure of the substitutability between two inputs. It can be calculated by: M σij =

fj Fjj fj Fij − , xi F xj F

(2.25)

where Fij and F are defined as above. In contrast to the direct elasticity of substitution and the Allen elasticity of substitution, the Morishima elasticity of substitution is usually not symmetric 1

The exponent of (−1) usually is the sum of the number of the deleted row (i + 1) and the number of the deleted column (j + 1), i.e. i + j + 2. In our case, we can simplify this to i + j, because (−1)i+j+2 = (−1)i+j · (−1)2 = (−1)i+j .

44

2 Primal Approach: Production Function M 6= σ M ). (σij ji

From the above definition of the Morishima elasticities of substitution (2.25), we can derive the relationship between the Morishima elasticities of substitution and the Allen elasticities of substitution: M σij

P

fj xj fj xj k fk xk Fij =P −P xi xj F k fk xk k fk xk fj xj fj xj =P σij − P σjj k fk xk k fk xk fj xj =P (σij − σjj ) , k fk xk

P

k fk x2j

xk Fjj F

(2.26) (2.27) (2.28)

where σjj can be calculated as the Allen elasticities of substitution with equation (2.21), but does not have an economic meaning. In case of two inputs (x = (x1 , x2 )), the Morishima elasticities of substitution are symmetric and equal to the direct and Allen elasticities of substitution: M σ12 =

= = = = =

f2 F12 f2 F22 − x1 F x2 F   F12 f2 f2 F22 − F x1 x2 F12 ! f2 f12 F12 f2 + F x1 x2 f1 f2   F12 f2 f1 + F x x2  1  F12 f2 x2 f1 x1 + F x1 x2 x1 x2 f1 x1 + f2 x2 F12 x1 x2 F

D = σ12 = σ12 .

(2.29) (2.30) (2.31) (2.32) (2.33) (2.34) (2.35)

2.1.10 Profit maximization We assume that the firms maximize their profit. The firm’s profit is given by: π=py−

X

wi xi ,

(2.36)

i

where p is the price of the output and wi is the price of the ith input. If the firm faces output price p and input prices wi , we can calculate the maximum profit that can be obtained by the firm by solving following optimization problem: max p y − y,x

X

wi xi , s.t. y = f (x)

i

45

(2.37)

2 Primal Approach: Production Function This restricted maximization can be transformed into an unrestricted optimization by replacing y by the production function: max p f (x) − x

X

wi x i

(2.38)

i

Hence, the first-order conditions are: ∂π ∂f (x) =p − wi = p M Pi − wi = 0 ∂xi ∂xi

(2.39)

wi = p M Pi = M V Pi

(2.40)

so that we get:

where M V Pi = p (∂y/∂xi ) is the marginal value product of the ith input.

2.1.11 Cost minimization Now, we assume that the firms take total output as given (e.g. because production is restricted by a quota) and try to produce this output quantity with minimal costs. The total cost is given by: c=

X

wi x i ,

(2.41)

i

where wi is the price of the ith input. If the firm faces input prices wi and wants to produce y units of output, the minimum costs can be obtained by: min x

X

wi xi , s.t. y = f (x)

(2.42)

i

This restricted minimization can be solved by using the Lagrangian approach: L=

X

wi xi + λ (y − f (x))

(2.43)

i

So that the first-order conditions are: ∂L ∂f (x) = wi − λ = wi − λ M Pi = 0 ∂xi ∂xi ∂L = y − f (x) = 0 ∂λ

(2.44) (2.45)

From the first-order conditions (2.44), we get:

and

wi = λM Pi

(2.46)

wi λM Pi M Pi = = = −M RT Sji wj λM Pj M Pj

(2.47)

46

2 Primal Approach: Production Function As profit maximization implies producing the optimal output quantity with minimum costs, the first-order conditions for the optimal input combinations (2.47) can be obtained not only from cost minimization but also from the first-order conditions for profit maximization (2.40): wi M V Pi p M Pi M Pi = = = = −M RT Sji wj M V Pj p M Pj M Pj

(2.48)

Equations (2.47) and (2.48) can also be expressed in terms of the RMRTS instead of the MRTS: wi x i xi wi x i wi x i ∂xj xi si = −M RT Sji = = = wjcxj = , − RM RT Sji = − ∂xi xj xj wj x j wj x j sj c

(2.49)

where si = wi xi /c are the cost shares. Hence, for a profit maximising or cost minimizing producer, the ratio of the cost shares of two inputs must be equal to the negative value of the (reversed) relative marginal rate of technical substitution (RMRTS). As the cost shares sum up to one, i.e.,

PN

i=1 si

= 1, we can revise equation (2.49) to:

si = −RM RT Sji sj =

i sj j



(2.50)



X i  1− = sk  j k∈{1,...,N }\j 

(2.51) 

X k  i = 1 − si j  k∈{1,...,N }\j i 

si i = 1 − j i = si +

si j

X k∈{1,...,N }\j



1 si 1 + j

k = 

X

k  =

k∈{1,...,N }\j

si = =

i si − j j

(2.52)

 X

k 

(2.53)

k∈{1,...,N }\j

X

k

(2.54)

k∈{1,...,N }\j

i j

(2.55)

i j

(2.56) i 

 P j 1 + 1j k∈{1,...,N }\j k

i P

j + i

k∈{1,...,N }\j k

(2.57) (2.58)

= PN

(2.59)

i 

(2.60)

=

k=1 k

Hence, for a profit maximising or cost minimizing producer, the the cost share of each input must

47

2 Primal Approach: Production Function be equal to the output elasticity of this input divided by the elasticity of scale.

2.1.12 Derived input demand functions and output supply functions In this section, we will analyze how profit maximizing or cost minimizing firms react on changing prices and on changing output quantities. 2.1.12.1 Derived from profit maximization If we replace the marginal products in the first-order conditions for profit maximization (2.40) by the equations for calculating these marginal products and then solve this system of equations for the input quantities, we get the input demand functions: xi = xi (p, w),

(2.61)

where w = [wi ] is the vector of all input prices. The input demand functions indicate the optimal input quantities (xi ) given the output price (p) and all input prices (w). We can obtain the output supply function from the production function by replacing all input quantities by the corresponding input demand functions: y = f (x(p, w)) = y(p, w),

(2.62)

where x(p, w) = [xi (p, w)] is the set of all input demand functions. The output supply function indicates the optimal output quantity (y) given the output price (p) and all input prices (w). Hence, the input demand and output supply functions can be used to analyze the effects of prices on the (optimal) input use and output supply. In economics, the effects of price changes are usually measured in terms of price elasticities. These price elasticities can measure the effects of the input prices on the input quantities: ij (p, w) =

∂xi (p, w) wj , ∂wj xi (p, w)

(2.63)

the effects of the input prices on the output quantity (expected to be non-positive): yj (p, w) =

∂y(p, w) wj , ∂wj y(p, w)

(2.64)

the effects of the output price on the input quantities (expected to be non-negative): ip (p, w) =

∂xi (p, w) p , ∂p xi (p, w)

48

(2.65)

2 Primal Approach: Production Function and the effect of the output price on the output quantity (expected to be non-negative): yp (p, w) =

p ∂y(p, w) . ∂p y(p, w)

(2.66)

The effect of an input price on the optimal quantity of the same input is expected to be nonpositive (ii (p, w) ≤ 0). If the cross-price elasticities between two inputs i and j are positive (ij (p, w) ≥ 0, ji (p, w) ≥ 0), they are considered as gross substitutes. If the cross-price elasticities between two inputs i and j are negative (ij (p, w) ≤ 0, ji (p, w) ≤ 0), they are considered as gross complements. 2.1.12.2 Derived from cost minimization If we replace the marginal products in the first-order conditions for cost minimization (2.46) by the equations for calculating these marginal products and the solve this system of equations for the input quantities, we get the conditional input demand functions: xi = xi (w, y)

(2.67)

These input demand functions are called “conditional,” because they indicate the optimal input quantities (xi ) given all input prices (w) and conditional on the fixed output quantity (y). The conditional input demand functions can be used to analyze the effects of input prices on the (optimal) input use if the output quantity is given. The effects of price changes on the optimal input quantities can be measured by conditional price elasticities: ij (w, y) =

∂xi (w, y) wj ∂wj xi (w, y)

(2.68)

The effect of the output quantity on the optimal input quantities can also be measured in terms of elasticities (expected to be positive): iy (w, y) =

∂xi (w, y) y . ∂y xi (w, y)

(2.69)

The conditional effect of an input price on the optimal quantity of the same input is expected to be non-positive (ii (w, y) ≤ 0). If the conditional cross-price elasticities between two inputs i and j are positive (ij (w, y) ≥ 0, ji (w, y) ≥ 0), they are considered as net substitutes. If the conditional cross-price elasticities between two inputs i and j are negative (ij (w, y) ≤ 0, ji (w, y) ≤ 0), they are considered as net complements.

2.1.13 Suitability of the production function for econometric applications Given the microeconomic theory of the production function and the assumptions of standard econometric methods such as ordinary least squares (OLS), it is appropriate to estimate a pro-

49

2 Primal Approach: Production Function duction function with these econometric methods if several conditions are fulfilled. The most important and relevant conditions are: 1. The inputs and outputs are rather similar across all firms in the data set (within each input or output category). 2. The production conditions are rather similar for all firms in the data set (unless the empirical specification appropriately accounts for these differences). 3. All firms in the data set produce the maximum obtainable output quantity given the input quantities. 4. No firm in the data set has all input quantities equal to zero but produces a strictly positive output quantity (i.e., strict essentiality must be fulfilled at all observations). 5. All input quantities (i.e., all explanatory variables) are uncorrelated with the error term, e.g., unobserved heterogeneity between firms (e.g., differences in management quality or production conditions such as soil quality or weather in agricultural production) is unrelated to the input quantities. This assumption is usually not fulfilled if the output quantity is exogenously given so that producers must adjust the input quantities to account for unobserved heterogeneity.

2.2 Productivity measures 2.2.1 Average products We calculate the average products of the three inputs for each firm in the data set by equation 2.2: > dat$apCap dat$apLab dat$apMat hist( dat$apCap ) > hist( dat$apLab ) > hist( dat$apMat ) The resulting graphs are shown in figure 2.1. These graphs show that average products (partial productivities) vary considerably between firms. Most firms in our data set produce on average between 0 and 40 units of output per unit of capital, between 2 and 16 units of output per unit of labor, and between 0 and 100 units of output per unit of materials. Looking at each average product separately, There are usually many firms with medium to low partial productivity measures and only a few firms with high partial productivity measures. We can find the firms with the highest partial productivities with the commands:

50

50

100

150

50 40 30 10 0

0

10 0 0

20

Frequency

15 5

10

Frequency

40 30 20

Frequency

50

20

60

2 Primal Approach: Production Function

0

5

10

apCap

15

20

25

apLab

0

50

150

250

350

apMat

Figure 2.1: Average products > which.max( dat$apCap ) [1] 132 > which.max( dat$apLab ) [1] 7 > which.max( dat$apMat ) [1] 83 Firm number 132 has the highest capital productivity, firm number 7 has the highest labor productivity, and firm number 83 has the highest materials productivity. The relationships between the average products can be visualized by scatter plots: > plot( dat$apCap, dat$apLab ) > plot( dat$apCap, dat$apMat ) > plot( dat$apLab, dat$apMat ) The resulting graphs are shown in figure 2.2. They show that the average products of the three inputs are positively correlated. As the units of measurements of the input and output quantities in our data set cannot be interpreted in practical terms, the interpretation of the size of the average products is practically not useful. However, they can be used to make comparisons between firms. For instance, the interrelation between average products and firm size can be analyzed. A possible (although not perfect) measure of size of the firms in our data set is the total output. > plot( dat$qOut, dat$apCap, log = "x" ) > plot( dat$qOut, dat$apLab, log = "x" ) > plot( dat$qOut, dat$apMat, log = "x" )

51

0

50

100

300 200 50

100

dat$apMat

200

150

0

0

0

50

100

dat$apMat

15 10 5

dat$apLab

20

25

300

2 Primal Approach: Production Function

0

dat$apCap

50

100

150

0

dat$apCap

5

10

15

20

dat$apLab

300 100 0

50

5 0 1e+05

5e+05

5e+06

200

apMat

15 10

apLab

100 50 0

apCap

20

25

150

Figure 2.2: Relationships between average products

1e+05

5e+05

qOut

5e+06

1e+05

qOut

Figure 2.3: Average products for different firm sizes

52

5e+05

5e+06 qOut

25

2 Primal Approach: Production Function The resulting graphs are shown in figure 2.3. These graphs show that the larger firms (i.e. firms with larger output quantities) produce also a larger output quantity per unit of each input. This is not really surprising, because the output quantity is in the numerator of equation (2.2) so that the average products are necessarily positively related to the output quantity for a given input quantity.

2.2.2 Total factor productivity After calculating a quantity index of all inputs (see section 1.5.1), we can use equation 2.3 to calculate the total factor productivity, where we arbitrarily choose the Fisher quantity index: > dat$tfp hist( dat$tfp )

0e+00

2e+06

4e+06

6e+06

0

TFP

1

2

3

4

5

dat$X

Figure 2.4: Total factor productivities The resulting histogram is shown in the left panel of figure 2.4. As the total factor productivity is the ratio between the aggregate output quantity and the aggregate input quantity, it can be illustrated by a scatter plot between the aggregate input quantity and the aggregate output quantity, whereas the slope of a line from the origin to a respective point in the scatter plot indicates the total factor productivity. The following commands create a scatter plot between the aggregate input quantity and the aggregate output quantity and add a line through the origin with the slope equal to the maximum total factor productivity as well as a line through the origin with the slope equal to the minimum total factor productivity in the data set: > plot( dat$X, dat$qOut, +

xlim = c( 0, max( dat$X ) ), ylim = c( 0, max( dat$qOut ) ) )

53

2 Primal Approach: Production Function > abline( 0, max( dat$tfp ) ) > abline( 0, min( dat$tfp ) ) The resulting scatter plot is shown in the right panel of figure 2.4. Both parts of figure 2.4 indicate that total factor productivity considerably varies between firms. Where do these large differences in (total factor) productivity come from? We can check the relation between total factor productivity and firm size by a scatter plot. We use two different measures of firm size: total output and aggregate input. The following commands produce scatter plots, where we set argument log of the plot command to the character string "x" so that the horizontal axis is measured in logarithmic terms and the dots (firms) are more equally spread: > plot( dat$qOut, dat$tfp, log = "x" )

6e+06 0e+00

3e+06

dat$tfp

3e+06 0e+00

dat$tfp

6e+06

> plot( dat$X, dat$tfp, log = "x" )

1e+05

5e+05

2e+06

1e+07

0.5

1.0

2.0

5.0

Figure 2.5: Firm size and total factor productivity The resulting scatter plots are shown in figure 2.5. The scatter plot in the left panel clearly shows that the firms with larger output quantities also have a larger total factor productivity. This is not really surprising, because the output quantity is in the numerator of equation (2.3) so that the total factor productivity is necessarily positively related to the output quantity for given input quantities. The total factor productivity is only slightly positively related to the measure of aggregate input use. We can also analyze whether the firms that use an advisory service have a higher total factor productivity than firms that do not use an advisory service. We can visualize and compare the total factor productivities of the two different groups of firms (with and without advisory service) using boxplot diagrams: > boxplot( tfp ~ adv, data = dat ) > boxplot( log(qOut) ~ adv, data = dat ) > boxplot( log(X) ~ adv, data = dat ) The resulting boxplot graphic is shown on the left panel of figure 2.6. It suggests that the firms

54

1.5 −0.5

0.0

log(X)

0.5

15

0e+00

−1.0

12

1e+06

13

14

log(qOut)

4e+06 3e+06 2e+06

TFP

5e+06

1.0

16

6e+06

17

7e+06

2 Primal Approach: Production Function

no advisory

advisory

no advisory

advisory

no advisory

advisory

Figure 2.6: Total factor productivities and advisory service that use advisory service are slightly more productive than firms that do not use advisory service (at least when looking at the 25th percentile and the median). However, these boxplots can only indicate a relationship between using advisory service and total factor productivity but they cannot indicate whether using an advisory service increases productivity (i.e. a causal effect). For instance, if larger firms are more likely to use an advisory service than smaller firms and larger firms have a higher total factor productivity than smaller firms, we expect that firms that use an advisory service have a higher productivity than smaller firms even if using an advisory service does not affect total factor productivity. However, this is not the case in our data set, because farms with and without advisory service use rather similar input quantities (see right panel of figure 2.6). As farms that use advisory service use similar input quantities but have a higher total factor productivity than farms without advisory service (see left panel of figure 2.6), they also have larger output quantities than corresponding farms without advisory service (see middle panel of figure 2.6). Furthermore, the causal effect of advisory service on total factor productivity might not be equal to the productivity difference between farms with and without advisory service, because it might be that the firms that anyway were the most productive were more (or less) likely to use advisory service than the firms that anyway were the least productive.

55

2 Primal Approach: Production Function

2.3 Linear production function 2.3.1 Specification A linear production function with N inputs is defined as: y = β0 +

N X

βi xi

(2.70)

i=1

2.3.2 Estimation We can add a stochastic error term to this linear production function and estimate it for our data set using the command lm: > prodLin summary( prodLin ) Call: lm(formula = qOut ~ qCap + qLab + qMat, data = dat) Residuals: Min

1Q

Median

3Q

Max

-3888955

-773002

86119

769073

7091521

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.616e+06

2.318e+05

-6.972 1.23e-10 ***

qCap

1.788e+00

1.995e+00

0.896

qLab

1.183e+01

1.272e+00

9.300 3.15e-16 ***

qMat

4.667e+01

1.123e+01

4.154 5.74e-05 ***

0.372

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1541000 on 136 degrees of freedom Multiple R-squared:

0.7868,

F-statistic: 167.3 on 3 and 136 DF,

Adjusted R-squared:

0.7821

p-value: < 2.2e-16

2.3.3 Properties As the coefficients of all three input quantities are positive, the monotonicity condition is (globally) fulfilled. However, the coefficient of the capital quantity is statistically not significantly different from zero. Therefore, we cannot be sure that the capital quantity has a positive effect

56

2 Primal Approach: Production Function on the output quantity. However, the result that the capital quantity does not have a statistically significant effect on the output quantity does not mean that the capital quantity in reality does not affect the output quantity—we just do not have sufficient (statistical) evidence to rule out that the capital quantity does not affect the output quantity. If the capital input in reality affects the output quantity but we remove it from our regression model, the estimation results will be biased and inconsistent. On the other hand, if the capital input in reality does not affect the output quantity but we keep it as explanatory variable in our regression model, the estimation results are inefficient but they are still unbiased and consistent. Furthermore, based on our knowledge about apple production, we expect that the capital input (consisting of land, fruit trees, machinery, . . . ) clearly affects the output quantity. For these reasons, we do not remove the capital quantity from the regression model but keep it as explanatory variable. As every linear function is concave (and convex), also our estimated linear production is concave and hence, also quasiconcave. As the isoquants of linear productions functions are linear, the input requirement sets are always convex (and concave). Our estimated linear production function does not fulfill the weak essentiality assumption, because the intercept is different from zero. The production technology described by a linear production function with more than one (relevant) input never shows strict essentiality. The input requirement sets derived from linear production functions are always closed and non-empty for y > 0 if weak essentiality is fulfilled (β0 = 0) and strict monotonicity is fulfilled for at least one input (∃ i ∈ {1, . . . , N } : βi > 0), as the input quantities must be non-negative (xi ≥ 0 ∀ i). The linear production function always returns finite, real, and single values for all non-negative and finite x. However, as the intercept of our estimated production function is negative, the nonnegativity assumption is not fulfilled. A linear production function would return non-negative values for all non-negative and finite x if β0 ≥ 0 and the monotonicity condition is fulfilled (βi ≥ 0 ∀ i = 1, . . . , N ). All linear production functions are continuous and twice-continuously differentiable.

2.3.4 Predicted output quantities We can calculate the predicted (“fitted”) output quantities manually by taking the linear production function (2.70), the observed input quantities, and the estimated parameters, but it is easier to use the fitted method to obtain the predicted values of the dependent variable from an estimated model: > dat$qOutLin all.equal( dat$qOutLin, coef( prodLin )[ "(Intercept)" ] + +

coef( prodLin )[ "qCap" ] * dat$qCap +

+

coef( prodLin )[ "qLab" ] * dat$qLab +

+

coef( prodLin )[ "qMat" ] * dat$qMat )

57

2 Primal Approach: Production Function [1] TRUE We can evaluate the “fit” of the model by comparing the observed with the fitted output quantities using the command compPlot (package miscTools): > library( "miscTools" ) > compPlot( dat$qOut, dat$qOutLin ) > compPlot( dat$qOut[ dat$qOutLin > 0 ], dat$qOutLin[ dat$qOutLin > 0 ],

fitted

2e+04

0.0e+00

2e+05

1.0e+07

fitted

2e+06

2e+07

log = "xy" )

2.0e+07

+

0.0e+00

1.0e+07

2.0e+07

2e+04

2e+05

observed

2e+06

2e+07

observed

Figure 2.7: Linear production function: fit of the model The resulting graphs are shown in figure 2.7. While the graph in the left panel uses a linear scale for the axes, the graph in the right panel uses a logarithmic scale for both axes. Hence, the deviations from the 45°-line illustrate the absolute deviations in the left panel and the relative deviations in the right panel. As the logarithm of non-positive values is undefined, we have to exclude observations with non-positive predicted output quantities in the graphs with logarithmic axes. The fit of the model looks okay in both scatter plots. As negative output quantities would render the corresponding output elasticities useless, we have to carefully check the sign of the predicted output quantities: > table( dat$qOutLin >= 0 ) FALSE

TRUE

1

139

One predicted output quantity is negative.

58

2 Primal Approach: Production Function

2.3.5 Marginal products In the linear production function, the marginal products are equal to coefficients of the corresponding input quantities. M Pi =

∂y = αi ∂xi

(2.71)

Hence, if a firm increases capital input by one unit, the output will increase by 1.79 units; if a firm increases labor input by one unit, the output will increase by 11.83 units; and if a firm increases materials input by one unit, the output will increase by 46.67 units.

2.3.6 Output elasticities As we do not know the units of measurements of the input and output quantities, the interpretation of the marginal products is practically not very useful. Therefore, we calculate the output elasticities (partial production elasticities) of the three inputs. i =

∂y xi xi M Pi = M Pi = ∂xi y y APi

(2.72)

As the output elasticities depend on the input and output quantities and these quantities generally differ between firms, also the output elasticities differ between firms. Hence, we can calculate them for each firm in the sample: > dat$eCap dat$eLab dat$eMat colMeans( subset( dat, , c( "eCap", "eLab", "eMat" ) ) ) eCap

eLab

eMat

0.1202721 2.0734793 0.8631936 However, these mean values are distorted by outliers (see figure 2.8). Therefore, we calculate the median values of the the output elasticities: > colMedians( subset( dat, , c( "eCap", "eLab", "eMat" ) ) ) eCap

eLab

eMat

0.08063406 1.28627208 0.58741460 Hence, if a firm increases capital input by one percent, the output will usually increase by around 0.08 percent; if the firm increases labor input by one percent, the output will often increase by around 1.29 percent; and if the firm increases materials input by one percent, the output will often increase by around 0.59 percent.

59

2 Primal Approach: Production Function We can visualize (the variation of) these output elasticities with histograms. The user can modify the desired number of bars in the histogram by adding an integer number as additional argument: > hist( dat$eCap ) > hist( dat$eLab, 20 )

0.4

0.8

1.2

10 0

10 0

0 0.0

20

Frequency

30

40 30 20

Frequency

60 40 20

Frequency

80

> hist( dat$eMat, 20 )

0

2

4

eCap

6

8

10 12 14

0

eLab

1

2

3

4

5

6

eMat

Figure 2.8: Linear production function: output elasticities The resulting graphs are shown in figure 2.8. If the firms increase capital input by one percent, the output of most firms will increase by between 0 and 0.2 percent; if the firms increase labor input by one percent, the output of most firms will increase by between 0.5 and 3 percent; and if the firms increase materials input by one percent, the output of most firms will increase by between 0.2 and 1.2 percent. While the marginal effect of capital on the output is rather small for most firms, there are many firms with implausibly high output elasticities of labor and materials (i > 1). This might indicate that the true production technology cannot be reasonably approximated by a linear production function. In contrast to a pure theoretical microeconomic model, our empirically estimated model includes a stochastic error term so that the observed output quantities (y) are not necessarily equal to the output quantities that are predicted by the model (ˆ y = f (x)). This error term comes from, e.g., measurement errors, omitted explanatory variables, (good or bad) luck, or unusual(ly) (good or bad) weather conditions. The better the fit of our model, i.e. the higher the R2 value, the smaller is the difference between the observed and the predicted output quantities. If we “believe” in our estimated model, it would be more consistent with microeconomic theory, if we use the predicted output quantities and disregard the stochastic error term. We can calculate the output elasticities based on the predicted output quantities (see section 2.3.4) rather than the observed output quantities: > dat$eCapFit dat$eLabFit dat$eMatFit colMeans( subset( dat, , c( "eCapFit", "eLabFit", "eMatFit" ) ) ) eCapFit

eLabFit

eMatFit

0.1421941 2.2142092 0.9784056 > colMedians( subset( dat, , c( "eCapFit", "eLabFit", "eMatFit" ) ) ) eCapFit

eLabFit

eMatFit

0.07407719 1.21044421 0.58821500 > hist( dat$eCapFit, 20 ) > hist( dat$eLabFit, 20 )

80 100 60 20 0

20 0

0 0.0 0.5 1.0 1.5 2.0 2.5 3.0

40

Frequency

120 80 60 40

Frequency

80 100 60 40 20

Frequency

> hist( dat$eMatFit, 20 )

−10

0

10

eCapFit

20

30

40

50

−5

0

eLabFit

5

10

15

20

25

eMatFit

Figure 2.9: Linear production function: output elasticities based on predicted output quantities The resulting graphs are shown in figure 2.9. While the choice of the variable for the output quantity (observed vs. predicted) only has a minor effect on the mean and median values of the output elasticities, the ranges of the output elasticities that are calculated from the predicted output quantities are much larger than the ranges of the output elasticities that are calculated from the observed output quantities. Due to 1 negative predicted output quantity, the output elasticities of this observation are also negative.

2.3.7 Elasticity of scale The elasticity of scale is the sum of all output elasticities =

X

i

i

Hence, the elasticities of scale of all firms in the sample can be calculated by:

61

(2.73)

2 Primal Approach: Production Function > dat$eScale dat$eScaleFit colMeans( subset( dat, , c( "eScale", "eScaleFit" ) ) ) eScale eScaleFit 3.056945

3.334809

> colMedians( subset( dat, , c( "eScale", "eScaleFit" ) ) ) eScale eScaleFit 1.941536

1.864253

Hence, if a firm increases all input quantities by one percent, the output quantity will usually increase by around 1.9 percent. This means that most firms have increasing returns to scale and hence, the firms could increase productivity by increasing the firm size (i.e. increasing all input quantities). The (variation of the) elasticities of scale can be visualized with histograms: > hist( dat$eScale, 30 ) > hist( dat$eScaleFit, 50 )

10

20

Frequency

30

60 20

40

Frequency

20 15 10 0

5

10 eScale

15

0

0

0

5

Frequency

25

40

30

> hist( dat$eScaleFit[ dat$eScaleFit > 0 & dat$eScaleFit < 15 ], 30 )

0

20

40

60

80

2

4

eScaleFit

6

8

10

12

14

0 < eScaleFit < 15

Figure 2.10: Linear production function: elasticities of scale The resulting graphs are shown in figure 2.10. As the predicted output quantity of 1 firm is negative, the elasticity of scale of this observation also is negative, if the predicted output quantities are used for the calculation. However, all remaining elasticities of scale that are based on the predicted output quantities are larger than one, which indicates increasing returns to scale. In contrast, 15 (out of 140) elasticities of scale that are calculated with the observed output quantities indicate decreasing returns to scale. However, both approaches indicate that most firms have

62

2 Primal Approach: Production Function an elasticity of scale between one and two. Hence, if these firms increase all input quantities by one percent, the output of most firms will increase by between 1 and 2 percent. Some firms even have an elasticity of scale larger than five, which is very implausible and might indicate that the true production technology cannot be reasonably approximated by a linear production function. Information on the optimal firm size can be obtained by analyzing the interrelationship between firm size and the elasticity of scale: > plot( dat$qOut, dat$eScale, log = "x" ) > abline( 1, 0 ) > plot( dat$X, dat$eScale, log = "x" ) > abline( 1, 0 ) > plot( dat$qOut, dat$eScaleFit, log = "x", ylim = c( 0, 15 ) ) > abline( 1, 0 ) > plot( dat$X, dat$eScaleFit, log = "x", ylim = c( 0, 15 ) )

15 5

10

eScale

10 5

eScale

15

> abline( 1, 0 )

0.5

1.0

2.0

5.0

1e+05

5e+05

5e+06

2e+07

5e+06

2e+07

10 0

5

eScaleFit 0

5

eScaleFit

10

15

qOut

15

X

2e+06

0.5

1.0

2.0

5.0

X

1e+05

5e+05

2e+06 qOut

Figure 2.11: Linear production function: elasticities of scale for different firm sizes

63

2 Primal Approach: Production Function The resulting graphs are shown in figure 2.11. They indicate that very small firms could enormously gain from increasing their size, while the benefits from increasing firm size decrease with size. Only a few elasticities of scale that are calculated with the observed output quantities indicate decreasing returns to scale so that productivity would decline when these firms increase their size. For all firms that use at least 2.1 times the input quantities of the average firm or produces more than 6,000,000 quantity units (approximately 6,000,000 Euros), the elasticities of scale that are based on the observed input quantities are very close to one. From this observation we could conclude that firms have their optimal size when they use at least 2.1 times the input quantities of the average firm or produce at least 6,000,000 quantity units (approximately 6,000,000 Euros turn over). In contrast, the elasticities of scale that are based on the predicted output quantities are larger one even for the largest firms in the data set. From this observation, we could conclude that the even the largest firms in the sample would gain from growing in size and thus, the most productive scale size is lager than the size of the largest firms in the sample. The high elasticities of scale explain why we found much higher partial productivities (average products) and total factor productivities for larger firms than for smaller firms.

2.3.8 Marginal rates of technical substitution As the marginal products based on a linear production function are equal to the coefficients, we can calculate the MRTS (2.17) as follows: > mrtsCapLab mrtsLabCap mrtsCapMat mrtsMatCap mrtsLabMat mrtsMatLab dat$rmrtsCapLab dat$rmrtsLabCap dat$rmrtsCapMat dat$rmrtsMatCap dat$rmrtsLabMat dat$rmrtsMatLab hist( dat$rmrtsCapLab, 20 ) > hist( dat$rmrtsLabCap ) > hist( dat$rmrtsCapMat ) > hist( dat$rmrtsMatCap ) > hist( dat$rmrtsLabMat ) > hist( dat$rmrtsMatLab ) The resulting graphs are shown in figure 2.12. According to the RMRTS based on the linear production function, most firms need between 20% more capital or around 2% more materials to compensate a 1% reduction of labor.

2.3.10 First-order conditions for profit maximization In this section, we will check to what extent the first-order conditions for profit maximization (2.40) are fulfilled, i.e. to what extent the firms use the optimal input quantities. We do this by comparing the marginal value products of the inputs with the corresponding input prices. We can calculate the marginal value products by multiplying the marginal products by the output price:

65

40 30 10

0

−0.20

0.00

−60

0

50 40

50

Frequency

30

0

0

10

10

20

Frequency rmrtsMatCap

0.0

−20

rmrtsCapMat

40

30 20 10 0 −0.5 −0.4 −0.3 −0.2 −0.1

−40

rmrtsLabCap

40

rmrtsCapLab

−0.10

30

−50

20

−100

0

0

0 −150

Frequency

20

Frequency

30

10

10

20

Frequency

40 30 20

Frequency

50

50

60

2 Primal Approach: Production Function

−1.5

−1.0

−0.5

rmrtsLabMat

0.0

−8

−6

−4

−2

0

rmrtsMatLab

Figure 2.12: Linear production function: relative marginal rates of technical substitution (RMRTS)

66

2 Primal Approach: Production Function > dat$mvpCap dat$mvpLab dat$mvpMat compPlot( dat$pCap, dat$mvpCap ) > compPlot( dat$pLab, dat$mvpLab ) > compPlot( dat$pMat, dat$mvpMat ) > compPlot( dat$pCap, dat$mvpCap, log = "xy" ) > compPlot( dat$pLab, dat$mvpLab, log = "xy" )

1

2

3

4

5

100 0 0

5

10 15 20 25 30 35

0

20

40

60

w Lab

80

120

w Mat

10

20

MVP Mat

5.0

5

0.5

0.2

1.0

0.5

2.0

1.0

MVP Lab

2.0

10.0

50

100

5.0

w Cap

MVP Cap

80 40 20

10 0

5

1 0 0

60

MVP Mat

25 20 15

MVP Lab

3 2

MVP Cap

4

30

5

35

140

> compPlot( dat$pMat, dat$mvpMat, log = "xy" )

0.2

0.5

1.0

2.0

5.0

0.5 1.0 2.0

w Cap

5.0

20.0

5

10

w Lab

20

50

100

w Mat

Figure 2.13: Marginal value products and corresponding input prices The resulting graphs are shown in figure 2.13. The graphs on the left side indicate that the marginal value products of capital are sometimes lower but more often higher than the capital prices. The four other graphs indicate that the marginal value products of labor and materials are always higher than the labor prices and the materials prices, respectively. This indicates that

67

2 Primal Approach: Production Function some firms could increase their profit by using more capital and all firms could increase their profit by using more labor and more materials. Given that most firms operate under increasing returns to scale, it is not surprising that most firms would gain from increasing most—or even all—input quantities. Therefore, the question arises why the firms in the sample did not do this. There are many possible reasons for not increasing the input quantities until the predicted optimal input levels, e.g. legal restrictions, environmental regulations, market imperfections, credit (liquidity) constraints, and/or risk aversion. Furthermore, market imperfections might cause that the (observed) average prices are lower than the marginal costs of obtaining these inputs (e.g. Henning and Henningsen, 2007), particularly for labor and capital.

2.3.11 First-order conditions for cost minimization As the marginal rates of technical substitution are constant for linear production functions, we compare the input price ratios with the negative inverse marginal rates of technical substitution by creating a histogram for each input price ratio and drawing a vertical line at the corresponding negative marginal rate of technical substitution: > hist( dat$pCap / dat$pLab ) > abline( v = - mrtsLabCap, lwd = 3

)

> hist( dat$pCap / dat$pMat ) > abline( v = - mrtsMatCap, lwd = 3

)

> hist( dat$pLab / dat$pMat ) > abline( v = - mrtsMatLab, lwd = 3

)

> hist( dat$pLab / dat$pCap ) > abline( v = - mrtsCapLab, lwd = 3

)

> hist( dat$pMat / dat$pCap ) > abline( v = - mrtsCapMat, lwd = 3

)

> hist( dat$pMat / dat$pLab ) > abline( v = - mrtsLabMat, lwd = 3

)

The resulting graphs are shown in figure 2.14. The upper left graph shows that the ratio between the capital price and the labor price is larger than the absolute value of the marginal rate of technical substitution between labor and capital (0.151) for the most firms in the sample: wcap M Pcap > −M RT Slab,cap = wlab M Plab

(2.74)

Or taken the other way round, the lower left graph shows that the ratio between the labor price and the capital price is smaller than the absolute value of the marginal rate of technical substitution between capital and labor (6.616) for the most firms in the sample: wlab M Plab < −M RT Scap,lab = wcap M Pcap

68

(2.75)

1

2

3

4

5

30 10 0

10 0

0 0

20

Frequency

30 20

Frequency

20 10

Frequency

30

40

40

40

50

2 Primal Approach: Production Function

0.0

0.2

0.6

0.8

0.1

w Cap / w Mat

0.2

0.3

0.4

w Lab / w Mat 40

2

4 w Lab / w Cap

6

8

30 10 0

10 0 0

20

Frequency

40 30 20

Frequency

40 20 0

Frequency

60

50

80

60

w Cap / w Lab

0.4

0

10

20

30

40

50

60

w Mat / w Cap

Figure 2.14: First-order conditions for costs minimization

69

5

10

15

w Mat / w Lab

20

2 Primal Approach: Production Function Hence, the firm can get closer to the minimum of the costs by substituting labor for capital, because this will decrease the marginal product of labor and increase the marginal product of capital so that the absolute value of the MRTS between labor and capital increases, the absolute value of the MRTS between capital and labor decreases, and both of the MRTS get closer to the corresponding input price ratios. Similarly, the graphs in the middle column indicate that almost all firms should substitute materials for capital and the graphs on the right indicate that most of the firms should substitute labor for materials. Hence, the firms could reduce production costs particularly by using less capital and more labor.

2.3.12 Derived input demand functions and output supply functions Given a linear production function (2.70), the input quantities chosen by a profit maximizing producer are either zero, indeterminate, or infinity:

xi (p, w) =

   0   

if M V Pi < wi

indeterminate

if M V Pi = wi

    ∞

(2.76)

if M V Pi > wi

If all input quantities are zero, the output quantity is equal to the intercept, which is zero in case of weak essentiality. Otherwise, the output quantity is indeterminate or infinity:

y(p, w) =

   β 0  

if M V Pi < wi ∀ i

∞

if M V Pi > wi ∃ i

    indeterminate

(2.77)

otherwise

A cost minimizing producer will use only a single input, i.e. the input with the lowest cost per unit of produced output (wi /M Pi ). If the lowest cost per unit of produced output can be obtained by two or more inputs, these input quantities are indeterminate.

xi (w, y) =

   0   

if

y−β0

βi     indeterminate

if

βi wi βi wi

< >

βj wj βj wj

∃j ∀ j 6= i

(2.78)

otherwise

Given that the unconditional and conditional input demand functions and the output supply functions based on the linear production function are non-continuous and often return either zero or infinite values, it does not make much sense to use this functional form to predict the effects of price changes when the true technology implies that firms always use non-zero finite input quantities.

70

2 Primal Approach: Production Function

2.4 Cobb-Douglas production function 2.4.1 Specification A Cobb-Douglas production function with N inputs is defined as: y=A

N Y xαi i .

(2.79)

i=1

This function can be linearized by taking the (natural) logarithm on both sides: ln y = α0 +

N X

αi ln xi ,

(2.80)

i=1

where α0 is equal to ln A.

2.4.2 Estimation We can estimate this Cobb-Douglas production function for our data set using the command lm: > prodCD summary( prodCD ) Call: lm(formula = log(qOut) ~ log(qCap) + log(qLab) + log(qMat), data = dat) Residuals: Min

1Q

Median

3Q

Max

-1.67239 -0.28024

0.00667

0.47834

1.30115

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -2.06377

1.31259

-1.572

0.1182

log(qCap)

0.16303

0.08721

1.869

log(qLab)

0.67622

0.15430

4.383 2.33e-05 ***

log(qMat)

0.62720

0.12587

4.983 1.87e-06 ***

0.0637 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.656 on 136 degrees of freedom Multiple R-squared:

0.5943,

F-statistic: 66.41 on 3 and 136 DF,

Adjusted R-squared: p-value: < 2.2e-16

71

0.5854

2 Primal Approach: Production Function

2.4.3 Properties The monotonicity condition is (globally) fulfilled, as the estimated coefficients of all three (logarithmic) input quantities are positive and the output quantity as well as all input quantities are non-negative (see equation 2.82). However, the coefficient of the (logarithmic) capital quantity is only statistically significantly different from zero at the 10% level. Therefore, we cannot be sure that the capital quantity has a positive effect on the output quantity.2 The quasiconcavity of our estimated Cobb-Douglas production function is checked in section 2.4.12. The production technology described by a Cobb-Douglas production function always shows weak and strict essentiality, because the output quantity becomes zero, as soon as a single input quantity becomes zero (see equation 2.79). The input requirement sets derived from Cobb-Douglas production functions are always closed and non-empty for y > 0 if strict monotonicity is fulfilled for at least one input (∃ i ∈ {1, . . . , N } : βi > 0), as the input quantities must be non-negative (xi ≥ 0 ∀ i). The Cobb-Douglas production function always returns finite, real, and single values if the input quantities are non-negative and finite. The predicted output quantity is non-negative as long as A and the input quantities are non-negative, where A = exp(α0 ) is positive even if α0 is negative. All Cobb-Douglas production functions are continuous and twice-continuously differentiable.

2.4.4 Predicted output quantities We can calculate the predicted (“fitted”) output quantities manually by taking the Cobb-Douglas function (2.79), the observed input quantities, and the estimated parameters, but it is easier to use the fitted method to obtain the predicted values of the dependent variable from an estimated model. As we estimated the Cobb-Douglas function in logarithms, we have to use the exponential function to obtain the predicted values in levels (non-logarithms): > dat$qOutCD all.equal( dat$qOutCD, +

with( dat, exp( coef( prodCD )[ "(Intercept)" ] ) *

+

qCap^coef( prodCD )[ "log(qCap)" ] *

+

qLab^coef( prodCD )[ "log(qLab)" ] *

+

qMat^coef( prodCD )[ "log(qMat)" ] ) )

[1] TRUE We can evaluate the “fit” of the Cobb-Douglas production function by comparing the observed with the fitted output quantities: 2

In section 2.3.3, we discuss whether input quantities that do not have a statistically significant effect on the output should be removed from the model.

72

2 Primal Approach: Production Function > compPlot( dat$qOut, dat$qOutCD )

1e+07 1e+05

5e+05 2e+06

fitted

1.0e+07 0.0e+00

fitted

2.0e+07

> compPlot( dat$qOut, dat$qOutCD, log = "xy" )

0.0e+00

1.0e+07

2.0e+07

1e+05

observed

5e+05

5e+06

observed

Figure 2.15: Cobb-Douglas production function: fit of the model The resulting graphs are shown in figure 2.15. While the graph in the left panel uses a linear scale for the axes, the graph in the right panel uses a logarithmic scale for both axes. Hence, the deviations from the 45°-line illustrate the absolute deviations in the left panel and the relative deviations in the right panel. The fit of the model looks okay in the scatter plot on the left-hand side, but if we use a logarithmic scale on both axes (as in the graph on the right-hand side), we can see that the output quantity is generally over-estimated if the the observed output quantity is small.

2.4.5 Output elasticities In the Cobb-Douglas function, the output elasticities of the inputs are equal to the corresponding coefficients: i =

∂y xi ∂ ln y ∂y ∂ ln xi xi ∂ ln y y 1 xi ∂ ln y = = = = αi ∂xi y ∂ ln xi ∂ ln y ∂xi y ∂ ln xi 1 xi y ∂ ln xi

(2.81)

Hence, if a firm increases capital input by one percent, the output will increase by 0.16 percent; if a firm increases labor input by one percent, the output will increase by 0.68 percent; and if a firm increases materials input by one percent, the output will increase by 0.63 percent. The output elasticity of capital is somewhat larger and the output elasticity of labor is considerably smaller when estimated by a Cobb-Douglas production function than when estimated by a linear production function. Indeed, the output elasticities of all three inputs are in the reasonable range, i.e. between zero one one, now.

73

2 Primal Approach: Production Function

2.4.6 Marginal products The marginal products of the inputs in a Cobb-Douglas production function can be derived in different ways. One possibility is to derive the marginal products from equation (2.79): M Pi =

N Y y ∂y = αi x−1 A xαj i = αi i ∂xi x i j=1

(2.82)

Another possibility is to derive the marginal products by re-arranging equation (2.81): M Pi =

∂ ln y y y ∂y = = αi ∂xi ∂ ln xi xi xi

(2.83)

As the marginal products depend on the input and output quantities and these quantities generally differ between firms, the marginal products based on Cobb-Douglas production functions also differ between firms. Hence, we can calculate them for each firm in the sample: > dat$mpCapCD dat$mpLabCD dat$mpMatCD dat$mpCapCDFit dat$mpLabCDFit dat$mpMatCDFit summary( dat[ , c( "mpCapCD", "mpCapCDFit", "mpLabCD", "mpLabCDFit", +

"mpMatCD", "mpMatCDFit" ) ] ) mpCapCD

Min.

: 0.2358

mpCapCDFit Min.

: 0.8782

mpLabCD Min.

: 0.581

mpLabCDFit Min.

: 1.870

1st Qu.: 2.4923

1st Qu.: 2.5344

1st Qu.: 3.465

1st Qu.: 4.250

Median : 3.6158

Median : 3.4891

Median : 6.220

Median : 5.397

Mean

Mean

Mean

Mean

: 5.3207

: 4.4904

: 6.905

: 5.909

3rd Qu.: 6.7871

3rd Qu.: 5.4819

3rd Qu.: 9.738

3rd Qu.: 7.114

Max.

Max.

Max.

Max.

:24.9217

mpMatCD Min.

:

5.157

1st Qu.: 28.750

:14.8856

mpMatCDFit Min.

: 21.89

1st Qu.: 35.49

74

:17.329

:13.757

2 Primal Approach: Production Function Median : 49.850

Median : 45.16

Mean

Mean

: 56.851

: 47.45

3rd Qu.: 70.450

3rd Qu.: 56.60

Max.

Max.

:189.056

:110.13

Hence, if a firm increases capital input by one unit, the output will on average increase by around 5.32 or 4.49 units. If the firm increases labor input by one unit, the output will on average increase by around 6.91 or 5.91 units. And if the firm increases materials input by one unit, the output will on average increase by around 56.85 or 47.45 units. If the “noise” term mainly comprises measurement errors in the output quantity, the “true” output quantity is probably closer to the predicted output quantity than to the observed output quantity so that it seems to be more appropriate to calculate the marginal products based on the predicted output quantity than based on the observed output quantity. On the other hand, if the “noise” term mainly comprises the effects of omitted variables, the “true” output quantity is probably closer to the observed output quantity than to the predicted output quantity so that it seems to be more appropriate to calculate the marginal products based on the observed output quantity than based on the predicted output quantity. However, in the second case, the output quantities used in the calculations are generally unequal to those predicted by the estimated production function, while the derivation of the marginal products in equation (2.82) assumes that the output quantities y are equal to the right-hand side of the production function (2.79). Therefore, using observed (rather than predicted) output quantities can result in inconsistencies in microeconomic analyses. We can visualize (the variation of) these marginal products with histograms: > hist( dat$mpCapCD ) > hist( dat$mpLabCD ) > hist( dat$mpMatCD ) > hist( dat$mpCapCDFit ) > hist( dat$mpLabCDFit ) > hist( dat$mpMatCDFit ) The resulting graphs are shown in figure 2.16. If the firms increase capital input by one unit, the output of most firms will increase by between 0 and 8 units; if the firms increase labor input by one unit, the output of most firms will increase by between 2 and 12 units; and if the firms increase materials input by one unit, the output of most firms will increase by between 20 and 80 units. Not surprisingly, a comparison of these marginal effects with the marginal effects from the linear production function confirms the results from the comparison based on the output elasticities: the marginal products of capital are generally larger than the marginal product estimated by the linear production function and the marginal products of labor are generally smaller than the marginal product estimated by the linear production function, while the marginal products of

75

30 5

10

15

20

25

30 10 0

5 0

0 0

0

5

10

15

0

MP Lab (observed output)

50

100

150

200

MP Mat (observed output)

0

5

10

15

MP Cap (predicted output)

40 30 10

20

Frequency

20 15 10

0

0

0

5

10

20

30

40

Frequency

50

25

60

30

MP Cap (observed output)

Frequency

20

Frequency

20 15 10

Frequency

30 20 10

Frequency

40

25

50

2 Primal Approach: Production Function

2

4

6

8

10

12

14

MP Lab (predicted output)

20

40

60

80

100

120

MP Mat (predicted output)

Figure 2.16: Cobb-Douglas production function: marginal products fuel are (on average) rather similar to the marginal product estimated by the linear production function.

2.4.7 Elasticity of scale As the elasticity of scale is the sum of all output elasticities (see equation 2.73), we can calculate it simply by summing up all coefficients except for the intercept: > sum( coef( prodCD )[ -1 ] ) [1] 1.466442 Hence, if the firm increases all input quantities by one percent, output will increase by 1.47 percent. This means that the technology has strong increasing returns to scale. However, in contrast to the results of the linear production function, the elasticity of scale based on the CobbDouglas production function is (globally) constant. Hence, it does not decrease (or increase), e.g., with the size of the firm. This means that the optimal firm size would be infinity. We can use the delta method (see section 1.5.4) to calculate the variance and the standard error of the elasticity of scale. Given that the first derivatives of the elasticity of scale with respect to the estimated coefficients are ∂ε/∂α0 = 0 and ∂ε/∂αCap = ∂ε/∂αLab = ∂ε/∂αM at = 1, we can do this by following commands:

76

2 Primal Approach: Production Function > ESCD dESCD varESCD seESCD tESCD cvESCD pESCD c( ESCD - cvESCD * seESCD, ESCD + cvESCD * seESCD ) [1] 1.251409 1.681476

77

2 Primal Approach: Production Function

2.4.8 Marginal rates of technical substitution The MRTS based on the Cobb-Douglas production function differ between firms. They can be calculated as follows: > dat$mrtsCapLabCD dat$mrtsLabCapCD dat$mrtsCapMatCD dat$mrtsMatCapCD dat$mrtsLabMatCD dat$mrtsMatLabCD all.equal( dat$mrtsCapLabCD, - dat$mpLabCDFit / dat$mpCapCDFit ) [1] TRUE > all.equal( dat$mrtsLabCapCD, - dat$mpCapCDFit / dat$mpLabCDFit ) [1] TRUE > all.equal( dat$mrtsCapMatCD, - dat$mpMatCDFit / dat$mpCapCDFit ) [1] TRUE > all.equal( dat$mrtsMatCapCD, - dat$mpCapCDFit / dat$mpMatCDFit ) [1] TRUE > all.equal( dat$mrtsLabMatCD, - dat$mpMatCDFit / dat$mpLabCDFit ) [1] TRUE > all.equal( dat$mrtsMatLabCD, - dat$mpLabCDFit / dat$mpMatCDFit ) [1] TRUE We can visualize (the variation of) these MRTSs with histograms: > hist( dat$mrtsCapLabCD ) > hist( dat$mrtsLabCapCD ) > hist( dat$mrtsCapMatCD ) > hist( dat$mrtsMatCapCD ) > hist( dat$mrtsLabMatCD ) > hist( dat$mrtsMatLabCD )

78

−3

−2

−1

0

30 10 −6

−4

−3

−2

−1

0

−50

mrtsMatCapCD

0.0

−10

0

50 Frequency

10 0

10 0 −0.2

−20

40

60 40 30 20

Frequency −0.4

−30

mrtsCapMatCD

50

50 40 30 20 10 0 −0.6

−40

mrtsLabCapCD

60

mrtsCapLabCD

−5

30

−4

20

−5

0

10 0

5 0 −6

Frequency

20

Frequency

40 30 20

Frequency

20 15 10

Frequency

25

50

30

60

35

2 Primal Approach: Production Function

−35

−25

−15

−5

mrtsLabMatCD

0

−0.4

−0.3

−0.2

−0.1

0.0

mrtsMatLabCD

Figure 2.17: Cobb-Douglas production function: marginal rates of technical substitution (MRTS) The resulting graphs are shown in figure 2.17. According to the MRTS based on the CobbDouglas production function, most firms only need between 0.5 and 2 additional units of capital or between 0.05 and 0.15 additional units of materials to replace one unit of labor.

2.4.9 Relative marginal rates of technical substitution As we do not know the units of measurements of the input quantities, the interpretation of the MRTSs is practically not very useful. To overcome this problem, we calculate the relative marginal rates of technical substitution (RMRTS) by equation (2.18). As the output elasticities based on a Cobb-Douglas production function are equal to the coefficients, we can calculate the RMRTS as follows: > rmrtsCapLabCD rmrtsLabCapCD rmrtsCapMatCD rmrtsMatCapCD rmrtsLabMatCD rmrtsMatLabCD dat$fCap dat$fLab dat$fMat dat$fCapCap dat$fLabLab dat$fMatMat dat$fCapLab dat$fCapMat dat$fLabMat dat$esdCapLab dat$esdCapMat dat$esdLabMat range( dat$esdCapLab ) [1] 1 1 > range( dat$esdCapMat ) [1] 1 1 > range( dat$esdLabMat ) [1] 1 1 All direct elasticities of substitution are exactly one for all observations and for all combinations of the inputs. This is no surprise and confirms that our calculations have been done correctly, because the Cobb-Douglas functional form implies that the direct elasticities of substitution are always equal to one, irrespective of the input and output quantities and the estimated parameters. 2.4.11.2 Allen elasticities of substitution In order to calculate the Allen elasticities of substitution, we need to construct the bordered Hessian matrix. As the first and second derivatives of the Cobb-Douglas function differ between observations, also the bordered Hessian matrix differs between observations. As a starting point, we construct the bordered Hessian Matrix just for the first observation: > bhm bhm[ 1, 2 ] dat$esmMatLab range( dat$esmMatLab )

86

2 Primal Approach: Production Function [1] 1 1 As with the direct elasticities of substitution and the Allen elasticities of substitution, the CobbDouglas functional form implies that all Morishima elasticities of substitution are always equal to one, irrespective of the input and output quantities and the estimated parameters. Using condition 2.24, we can show that all Morishima elasticities of substitution are always M = 1 ∀ i 6= j), if all Allen elasticities of substitution are one (σ = 1 ∀ i 6= j): one (σij ij M σij = Kj σij − Kj σjj = Kj +

X k6=j

Kk σkj =

X

Kk = 1

(2.96)

k

Hence, the Cobb-Douglas function cannot be used to analyze the substitutability of the inputs, because it will always return (direct, Allen and Morishima) elasticities of substitution that are equal to one, even if the true elasticities of substitution are close to zero or close to infinity.

2.4.12 Quasiconcavity We start by checking whether our estimated Cobb-Douglas production function is quasiconcave at the first observation: > bhm [,1]

[,2]

[1,]

0.000000

6.229014e+00

[2,]

6.229014 -6.202845e-05

[3,]

6.031225

[4,] 59.090913

[,3]

[,4]

6.031225e+00 59.0909133861 1.169835e-05

0.0001146146

1.169835e-05 -5.423455e-06

0.0001109752

1.146146e-04

1.109752e-04 -0.0006462733

> det( bhm[ 1:2, 1:2 ] ) [1] -38.80062 > det( bhm[ 1:3, 1:3 ] ) [1] 0.003345742 > det( bhm ) [1] -1.013458e-05 The first leading principal minor of the bordered Hessian matrix is negative, the second principal minor is positive, and the third principal minor is negative. This means that our estimated Cobb-Douglas production function is quasiconcave at the first observation. Now we check quasiconcavity at each observation based on the three-dimensional array that consists of the stacked bordered Hessian matrices of all observations:

87

2 Primal Approach: Production Function > dat$quasiConc 0 &

+

apply( bhmArray, 3, det ) < 0

> table( dat$quasiConc ) TRUE 140 Our estimated Cobb-Douglas production function is quasiconcave at all of the 140 observations. In fact, all Cobb-Douglas production functions are quasiconcave in inputs if A ≥ 0, α1 ≥ 0, . . . , αN ≥ 0, while Cobb-Douglas production functions are concave in inputs if A ≥ 0, α1 ≥ 0, PN

. . . , αN ≥ 0, and the technology has decreasing or constant returns to scale (

i=1 αi

≤ 1).3

2.4.13 First-order conditions for profit maximization In this section, we will check to what extent the first-order conditions (2.40) for profit maximization are fulfilled, i.e. to what extent the firms use the optimal input quantities. We do this by comparing the marginal value products of the inputs with the corresponding input prices. We can calculate the marginal value products by multiplying the marginal products by the output price: > dat$mvpCapCd dat$mvpLabCd dat$mvpMatCd compPlot( dat$pCap, dat$mvpCapCd ) > compPlot( dat$pLab, dat$mvpLabCd ) > compPlot( dat$pMat, dat$mvpMatCd ) > compPlot( dat$pCap, dat$mvpCapCd, log = "xy" ) > compPlot( dat$pLab, dat$mvpLabCd, log = "xy" ) > compPlot( dat$pMat, dat$mvpMatCd, log = "xy" ) The resulting graphs are shown in figure 2.18. They indicate that the marginal value products are always nearly equal to or higher than the corresponding input prices. This indicates that (almost) all firms could increase their profit by using more of all inputs. Given that the estimated Cobb-Douglas technology exhibits increasing returns to scale, it is not surprising that (almost) all firms would gain from increasing all input quantities. Therefore, the question arises why the firms in the sample did not do this. This questions has already been addressed in section 2.3.10. 3

See, e.g., http://econren.weebly.com/uploads/9/0/1/5/9015734/lecture16.pdf or http://web.mit.edu/14. 102/www/ps/ps1sol.pdf.

88

30

50 0 0

5

10

20

25

30

0

100

w Lab

MVP Mat

10.0 5.0

200

250

0.5

5

1.0

10

2.0

MVP Lab

150

w Mat

20.0 5.0 2.0 0.5 0.2

MVP Cap

50

100 200

w Cap

15

50

20

20

10

150

MVP Mat

5 0

0 0

100

25 20 10

15

MVP Lab

30 20 10

MVP Cap

200

30

250

2 Primal Approach: Production Function

0.2

0.5

2.0

5.0

20.0

0.5

1.0

2.0

w Cap

5.0

20.0

5

10

w Lab

Figure 2.18: Marginal value products and corresponding input prices

89

20 w Mat

50

200

2 Primal Approach: Production Function

2.4.14 First-order conditions for cost minimization As the marginal rates of technical substitution differ between observations for the Cobb-Douglas functional form, we use scatter plots for visualizing the comparison of the input price ratios with the negative inverse marginal rates of technical substitution: > compPlot( dat$pCap / dat$pLab, - dat$mrtsLabCapCD ) > compPlot( dat$pCap / dat$pMat, - dat$mrtsMatCapCD ) > compPlot( dat$pLab / dat$pMat, - dat$mrtsMatLabCD ) > compPlot( dat$pCap / dat$pLab, - dat$mrtsLabCapCD, log = "xy" ) > compPlot( dat$pCap / dat$pMat, - dat$mrtsMatCapCD, log = "xy" )

0.3 0.1

0.0

1 0

1

2

3

4

5

6

0.0

0.2

0.6

0.8

0.1

w Cap / w Mat

0.2

0.3

0.4

w Lab / w Mat

0.2

0.5

1.0

2.0

5.0

0.20

− MRTS Mat Lab

0.05

0.05 0.10 0.20 0.02

0.2

0.5

1.0

− MRTS Mat Cap

2.0

0.50

5.0

w Cap / w Lab

0.4

0.10

0

− MRTS Lab Cap

0.2

− MRTS Mat Lab

0.6 0.2

0.4

− MRTS Mat Cap

4 3 2

− MRTS Lab Cap

5

0.4

0.8

6

> compPlot( dat$pLab / dat$pMat, - dat$mrtsMatLabCD, log = "xy" )

0.02

w Cap / w Lab

0.05

0.20

0.50

0.05

w Cap / w Mat

0.10

0.20

w Lab / w Mat

Figure 2.19: First-order conditions for cost minimization The resulting graphs are shown in figure 2.19. Furthermore, we use histograms to visualize the (absolute and relative) differences between the input price ratios and the corresponding negative inverse marginal rates of technical substitution: > hist( - dat$mrtsLabCapCD - dat$pCap / dat$pLab ) > hist( - dat$mrtsMatCapCD - dat$pCap / dat$pMat )

90

2 Primal Approach: Production Function > hist( - dat$mrtsMatLabCD - dat$pLab / dat$pMat ) > hist( log( - dat$mrtsLabCapCD / ( dat$pCap / dat$pLab ) ) ) > hist( log( - dat$mrtsMatCapCD / ( dat$pCap / dat$pMat ) ) )

30 10

20

Frequency

40 30 20

Frequency

60 40 −4

−2

0

2

4

0

0

0

10

20

Frequency

80

50

40

> hist( log( - dat$mrtsMatLabCD / ( dat$pLab / dat$pMat ) ) )

−0.6

−0.2

0.0

0.2

−0.3 −0.2 −0.1

−MrtsMatCap − wCap / wMat

0.0

0.1

0.2

−MrtsMatLab − wLab / wMat

−1.0

0.0

1.0

log(−MrtsLabCap / (wCap / wLab))

15

Frequency

5 0

0 −2.0

10

40 30 10

20

Frequency

40 30 20 0

10

Frequency

20

50

−MrtsLabCap − wCap / wLab

−0.4

−2.5

−1.5

−0.5

0.5

log(−MrtsMatCap / (wCap / wMat))

−1.5

−0.5

0.0

0.5

1.0

log(−MrtsMatLab / (wLab / wMat))

Figure 2.20: First-order conditions for costs minimization The resulting graphs are shown in figure 2.20. The left graphs in figures 2.19 and 2.20 show that the ratio between the capital price and the labor price is larger than the absolute value of the marginal rate of technical substitution between labor and capital for the most firms in the sample: wcap M Pcap > −M RT Slab,cap = wlab M Plab

(2.97)

Hence, most firms can get closer to the minimum of their production costs by substituting labor for capital, because this will decrease the marginal product of labor and increase the marginal product of capital so that the absolute value of the MRTS between labor and capital increases and gets closer to the corresponding input price ratio. Similarly, the graphs in the middle column indicate that most firms should substitute materials for capital and the graphs on the right indicate that the majority of the firms should substitute materials for labor. Hence, the majority

91

2 Primal Approach: Production Function of the firms could reduce production costs particularly by using less capital and more materials4 but there might be (legal) regulations that restrict the use of materials (e.g. fertilizers, pesticides). Given that cost minimization (as well as profit maximisation) implies that the cost shares are equal to the corresponding output elasticities divided by the elasticity of scale (see equation 2.60), we can check the conditions for cost minimization also by: > hist( dat$sCap ) > abline( v = coef( prodCD )[ "log(qCap)" ] / sum( coef( prodCD )[-1] ), +

lwd = 2 )

> hist( dat$sLab ) > abline( v = coef( prodCD )[ "log(qLab)" ] / sum( coef( prodCD )[-1] ), +

lwd = 2 )

> hist( dat$sMat ) > abline( v = coef( prodCD )[ "log(qMat)" ] / sum( coef( prodCD )[-1] ),

25 20 10

15

Frequency

20 0.0

0.1

0.2

0.3

0.4

5 0

0

0

5

5

10

10

15

Frequency

20 15

Frequency

25

30

25

35

30

lwd = 2 ) 30

+

0.3

0.4

dat$sCap

0.5

0.6

dat$sLab

0.7

0.8

0.1

0.2

0.3

0.4

0.5

0.6

dat$sMat

Figure 2.21: Cost shares and output elasticities divided by the elasticity of scale The resulting graphs are shown in figure 2.21. They indicate that the majority of firms uses more capital and less material than would be cost-minimizing according to our estimated Cobb-Douglas production function.

4

This generally confirms the results of the linear production function for the relationships between capital and labor and the relationship between capital and materials. However, in contrast to the linear production function, the results obtained by the Cobb-Douglas functional form indicate that most firms should substitute materials for labor (rather than the other way round).

92

2 Primal Approach: Production Function

2.4.15 Derived input demand functions and output supply functions Given a Cobb-Douglas production function (2.79), the input quantities chosen by a profit maximizing producer are

xi (p, w) =

    Y  α  i  P A    wi

αj wj

j

1 !αj  1−α

if α < 1



(2.98)

  0∨∞      ∞

if α = 1 if α > 1

and the output quantity is

y(p, w) =

with α =

P

j

   Y    A P α     j

αj wj

1 !αj  1−α

if α < 1



 0∨∞        ∞

if α = 1

(2.99)

if α > 1

αj . Hence, if the Cobb-Douglas production function exhibits increasing returns

to scale ( = α > 1), the optimal input and output quantities are infinity. As our estimated Cobb-Douglas production function has increasing returns to scale, the optimal input quantities are infinity. Therefore, we cannot evaluate the effect of prices on the optimal input quantities. A cost minimizing producer would choose the following input quantities: 

y Y α i wj xi (w, y) =  A j6=i αj wi

!αj  α1

(2.100)



For our three-input Cobb-Douglas production function, we get following conditional input demand functions 

y xcap (w, y) =  A

αcap wcap

xlab (w, y) =

y A

wcap αcap

xmat (w, y) =

y A

wcap αcap

!αlab +αmat 

!αcap 

!αcap 

wlab αlab

αlab 

wmat αmat

αlab wlab

αcap +αmat 

wlab αlab

αlab 

αmat wmat

αmat

wmat αmat



1 αcap +αlab +αmat



(2.101)

1 αmat ! αcap +αlab +αmat

(2.102)

1 αcap +αlab ! αcap +αlab +αmat

(2.103)

We can use these formulas to calculate the cost-minimizing input quantities based on the observed input prices (w) and the predicted output quantities (f (x)). Alternatively, we could calculate the cost-minimizing input quantities based on the observed input prices (w) and the observed

93

2 Primal Approach: Production Function output quantities (y). However, in the latter case, the predicted output quantities based on the cost-minimizing input quantities would differ from the predicted output quantities based on the observed input quantities (i.e. y = f (x(w, y)) 6= f (x)) so that a comparison of the cost-minimizing input quantities (x(w, y)) with the observed input quantities (x) would be less useful. As the coefficients of the Cobb-Douglas function repeatedly occur in the formulas for calculating the cost-minimizing input quantities, it is convenient to define short-cuts for them: > A aCap aLab aMat dat$qCapCD dat$qLabCD dat$qMatCD dat$qOutTest all.equal( dat$qOutCD, dat$qOutTest ) [1] TRUE Given that the output quantities predicted from the cost-minimizing input quantities are all equal to the output quantities predicted from the observed input quantities, we can be pretty sure that our calculations are correct. Now, we can use scatter plots to compare the cost-minimizing input quantities with the observed input quantities: > compPlot( dat$qCapCD, dat$qCap ) > compPlot( dat$qLabCD, dat$qLab )

94

2 Primal Approach: Production Function > compPlot( dat$qMatCD, dat$qMat ) > compPlot( dat$qCapCD, dat$qCap, log = "xy" ) > compPlot( dat$qLabCD, dat$qLab, log = "xy" )

0e+00

2e+05

4e+05

100000 60000

qMat

600000

6e+05

20000

0e+00

200000

2e+05

qLab

qCap

4e+05

1000000

6e+05

> compPlot( dat$qMatCD, dat$qMat, log = "xy" )

200000

600000

20000

qLabCD

60000

100000

qMatCD

5e+03

2e+04

1e+05

5e+05

5e+04

5e+03

2e+04

qMat

5e+04

5e+05 5e+04

2e+05

qLab

1e+05 5e+03

2e+04

qCap

5e+05

qCapCD

1000000

2e+05

qCapCD

5e+05

5e+03

qLabCD

2e+04

5e+04

qMatCD

Figure 2.22: Optimal and observed input quantities The resulting graphs are shown in figure 2.22. As we already found out in section 2.4.14, many firms could reduce their costs by substituting materials for capital. We can also evaluate the potential for cost reductions by comparing the observed costs with the costs when using the cost-minimizing input quantities: > dat$costProdCD mean( dat$costProdCD / dat$cost ) [1] 0.9308039 Our model predicts that the firms could reduce their costs on average by 7% by using costminimizing input quantities. The variation of the firms’ cost reduction potentials are shown by a histogram:

95

2 Primal Approach: Production Function

15 0 5

Frequency

25

> hist( dat$costProdCD / dat$cost )

0.75

0.80

0.85

0.90

0.95

1.00

costProdCD / cost

Figure 2.23: Minimum total costs as share of actual total costs The resulting graph is shown in figure 2.23. While many firms have a rather small potential for reducing costs by reallocating input quantities, there are some firms that could save up to 25% of their total costs by using the optimal combination of input quantities. We can also compare the observed input quantities with the cost-minimizing input quantities and the observed costs with the minimum costs for each single observation (e.g. when consulting individual firms in the sample): > round( subset( dat, , c("qCap", "qCapCD", "qLab", "qLabCD", "qMat", "qMatCD", +

"cost", "costProdCD") ) )[1:5,] qCap qCapCD

qLab qLabCD

qMat qMatCD

cost costProdCD

1 84050

33720 360066 405349 34087

38038 846329

790968

2 39663

18431 249769 334442 40819

36365 580545

545777

3 37051

14257 140286 135701 24219

32176 306040

281401

4 21222

13300

83427

69713 18893

25890 199634

191709

5 44675

28400

89223 108761 14424

13107 226578

221302

2.4.16 Derived input demand elasticities We can measure the effect of the input prices and the output quantity on the cost-minimizing input quantities by calculating the conditional price elasticities based on the partial derivatives of the conditional input demand functions (2.100) with respect to the input prices and the output quantity. In case of two inputs, we can calculate the demand elasticities of the first input by: 1

y α1 w2 α2 α x1 (w, y) = A α2 w1 w1 ∂x1 (w, y) 11 (w, y) = ∂w1 x1 (w, y) 





(2.104) (2.105)

96

2 Primal Approach: Production Function

=

1 α



y A



1 =− α



1 =− α



α1 w2 α2 w1

y A



y A



1 y =− α A 1 α2 = − x1 α x1 ∂x1 (w, y) 12 (w, y) = ∂w2 

1 = α



1 = α



1 = α



y A



y A



y A



α2  1 −1 α

1 = α



α2  1 −1

α1 w2 α2 w1

α2  1 −1

α

α

1 = α





y A





y A



−

α 1 w2 α 2 w1

α2 −1

α 1 w2 α 2 w1

α2

α 1 w2 α2 w12



w1 x1

α1 w2 α2 α2 w1 x1

α2 x1

α1 w2 α2 w1

α2  1 −1

α1 w2 α2 w1

α2  1 −1

α1 w2 α2 w1

α2  1 −1

α

α

α



1

α

(2.107) (2.108)

y α1 w2 α2 A α2 w1 

y A



y A



(2.109) (2.110) (2.111)

α2 −1

α1 w2 α2 w1

α2 −1

α1 w2 α2 w1

 α2

α 1 1 w2 α 2 w1 x 1

α1 w2 α2 α2 w1 x1

α2 x1

α2 x1

(2.112) (2.113) (2.114) (2.115) (2.116) (2.117)

α1 w2 α2 w1

α2  1 −1

α1 w2 α2 w1

α2  1 −1



(2.106)

1



1 y α1 w2 = α A α2 w1 1 1 1 = x1 = α x1 α 

y A

α2 −1 

α1 w2 α2 α α2 α2 w1 x1 α2 α1 − α α1 =− = = −1 α α α w2 x1 (w, y)





y A



α1 w2 α2 w1

1 y α1 w2 α2 = α A α2 w1 α2 1 α2 = = x1 α x1 α ∂x1 (w, y) y 1y (w, y) = ∂y x1 (w, y) 

y α1 w2 α2 A α2 w1

α

α

 α2  1

α

1 A



y A



α1 w2 α2 w1

 α2

α1 w2 α2 w1

 α2

y x1

(2.118)

1 x1

(2.119)

1 x1

(2.120) (2.121)

and analogously the demand elasticities of the second input: 1

y α2 w1 α1 α x2 (w, y) = A α 1 w2 ∂x2 (w, y) w2 α1 α2 − α α2 22 (w, y) = =− = = −1 ∂w2 x2 (w, y) α α α ∂x2 (w, y) w1 α1 21 (w, y) = = ∂w1 x2 (w, y) α ∂x2 (w, y) y 1 2y (w, y) = = . ∂y x2 (w, y) α 





97

(2.122) (2.123) (2.124) (2.125)

2 Primal Approach: Production Function One can similarly derive the input demand elasticities for the general case of N inputs: ∂xi (w, y) wj αj = − δij ∂wj xi (w, y) α y ∂xi (w, y) 1 iy (w, y) = = , ∂y xi (w, y) α ij (w, y) =

(2.126) (2.127)

where δij is (again) Kronecker’s delta (2.95). We have calculated all these elasticities based on the estimated coefficients of the Cobb-Douglas production function; these elasticities are presented in table 2.1. If the price of capital increases by one percent, the cost-minimizing firm will decrease the use of capital by 0.89% and increase the use of labor and materials by 0.11% each. If the price of labor increases by one percent, the cost-minimizing firm will decrease the use of labor by 0.54% and increase the use of capital and materials by 0.46% each. If the price of materials increases by one percent, the cost-minimizing firm will decrease the use of materials by 0.57% and increase the use of capital and labor by 0.43% each. If the cost-minimizing firm increases the output quantity by one percent, (s)he will increase all input quantities by 0.68%. Table 2.1: Conditional demand elasticities wcap xcap -0.89 xlab 0.11 xmat 0.11

derived from Cobb-Douglas production function wlab wmat y 0.46 0.43 0.68 -0.54 0.43 0.68 0.46 -0.57 0.68

2.5 Quadratic production function 2.5.1 Specification A quadratic production function is defined as y = β0 +

X

βi xi +

i

1 XX βij xi xj , 2 i j

(2.128)

where the restriction βij = βji is required to identify all coefficients, because xi xj and xj xi are the same regressors. Based on this general form, we can derive the specification of a quadratic production function with three inputs: 1 1 1 y = β0 +β1 x1 +β2 x2 +β3 x3 + β11 x21 + β22 x22 + β33 x23 +β12 x1 x2 +β13 x1 x3 +β23 x2 x3 (2.129) 2 2 2

2.5.2 Estimation We can estimate this quadratic production function with the command > prodQuad summary( prodQuad ) Call: lm(formula = qOut ~ qCap + qLab + qMat + I(0.5 * qCap^2) + I(0.5 * qLab^2) + I(0.5 * qMat^2) + I(qCap * qLab) + I(qCap * qMat) + I(qLab * qMat), data = dat) Residuals: Min

1Q

Median

3Q

Max

-3928802

-695518

-186123

545509

4474143

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

-2.911e+05

3.615e+05

-0.805 0.422072

qCap

5.270e+00

4.403e+00

1.197 0.233532

qLab

6.077e+00

3.185e+00

1.908 0.058581 .

qMat

1.430e+01

2.406e+01

0.595 0.553168

I(0.5 * qCap^2)

5.032e-05

3.699e-05

1.360 0.176039

I(0.5 * qLab^2) -3.084e-05

2.081e-05

-1.482 0.140671

I(0.5 * qMat^2) -1.896e-03

8.951e-04

-2.118 0.036106 *

I(qCap * qLab)

-3.097e-05

1.498e-05

-2.067 0.040763 *

I(qCap * qMat)

-4.160e-05

1.474e-04

-0.282 0.778206

I(qLab * qMat)

4.011e-04

1.112e-04

3.608 0.000439 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 1344000 on 130 degrees of freedom Multiple R-squared:

0.8449,

F-statistic: 78.68 on 9 and 130 DF,

Adjusted R-squared:

0.8342

p-value: < 2.2e-16

Although many of the estimated coefficients are statistically not significantly different from zero, the statistical significance of some quadratic and interaction terms indicates that the linear production function, which neither has quadratic terms not interaction terms, is not suitable to model the true production technology. As the linear production function is “nested” in the quadratic production function, we can apply a “Wald test” or a likelihood ratio test to check whether the linear production function is rejected in favor of the quadratic production function. These tests can be done by the functions waldtest and lrtest (package lmtest): > library( "lmtest" )

99

2 Primal Approach: Production Function > waldtest( prodLin, prodQuad ) Wald test Model 1: qOut ~ qCap + qLab + qMat Model 2: qOut ~ qCap + qLab + qMat + I(0.5 * qCap^2) + I(0.5 * qLab^2) + I(0.5 * qMat^2) + I(qCap * qLab) + I(qCap * qMat) + I(qLab * qMat) Res.Df Df 1

136

2

130

F

Pr(>F)

6 8.1133 1.869e-07 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> lrtest( prodLin, prodQuad ) Likelihood ratio test Model 1: qOut ~ qCap + qLab + qMat Model 2: qOut ~ qCap + qLab + qMat + I(0.5 * qCap^2) + I(0.5 * qLab^2) + I(0.5 * qMat^2) + I(qCap * qLab) + I(qCap * qMat) + I(qLab * qMat) #Df

LogLik Df

1

5 -2191.3

2

11 -2169.1

Chisq Pr(>Chisq)

6 44.529

5.806e-08 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

These tests show that the linear production function is clearly inferior to the quadratic production function and hence, should not be used for analyzing the production technology of the firms in this data set.

2.5.3 Properties We cannot see from the estimated coefficients whether the monotonicity condition is fulfilled. Unless all coefficients are non-negative (but not necessarily the intercept), quadratic production functions cannot be globally monotone, because there will always be a set of input quantities that result in negative marginal products. We will check the monotonicity condition at each observation in section 2.5.5. Our estimated quadratic production function does not fulfill the weak essentiality assumption, because the intercept is different from zero (but its deviation from zero is not statistically signif-

100

2 Primal Approach: Production Function icant). The production technology described by a quadratic production function with more than one (relevant) input never shows strict essentiality. The input requirement sets derived from quadratic production functions are always closed and non-empty. The quadratic production function always returns finite, real, and single values but the nonnegativity assumption is only fulfilled, if all coefficients (including the intercept), are non-negative. All quadratic production functions are continuous and twice-continuously differentiable.

2.5.4 Predicted output quantities We can obtain the predicted output quantities with the fitted method: > dat$qOutQuad compPlot( dat$qOut, dat$qOutQuad )

0.0e+00

1e+07 1e+05

5e+05 2e+06

fitted

1.0e+07 0.0e+00

fitted

2.0e+07

> compPlot( dat$qOut, dat$qOutQuad, log = "xy" )

1.0e+07

2.0e+07

1e+05

observed

5e+05

5e+06

observed

Figure 2.24: Quadratic production function: fit of the model The resulting graphs are shown in figure 2.24. While the graph in the left panel uses a linear scale for the axes, the graph in the right panel uses a logarithmic scale for both axes. Hence, the deviations from the 45°-line illustrate the absolute deviations in the left panel and the relative deviations in the right panel. The fit of the model looks okay in the scatter plot on the left-hand side, but if we use a logarithmic scale on both axes (as in the graph on the right-hand side), we can see that the output quantity is over-estimated if the the observed output quantity is small. As negative output quantities would render the corresponding output elasticities useless, we have carefully check the sign of the predicted output quantities:

101

2 Primal Approach: Production Function > table( dat$qOutQuad >= 0 ) TRUE 140 Fortunately, not a single predicted output quantity is negative.

2.5.5 Marginal products In case of a quadratic production function, the marginal products are M Pi = β i +

X

βij xj

(2.130)

j

We can simplify the code for computing the marginal products and some other figures by using short names for the coefficients: > b1 b2 b3 b11 b22 b33 b12 table( dat$mpCapQuad >= 0 ) FALSE

TRUE

28

112

> table( dat$mpLabQuad >= 0 ) FALSE

TRUE

5

135

> table( dat$mpMatQuad >= 0 ) FALSE

TRUE

8

132

> dat$monoQuad = 0 & mpLabQuad >= 0 & mpMatQuad >= 0 ) > table( dat$monoQuad ) FALSE

TRUE

39

101

28 firms have a negative marginal product of capital, 5 firms have a negative marginal product of labor, and 8 firms have a negative marginal product of materials. In total the monotonicity condition is not fulfilled at 39 out of 140 observations. Although the monotonicity conditions are still fulfilled for the largest part of firms in our data set, these frequent violations could indicate a possible model misspecification.

103

2 Primal Approach: Production Function

2.5.6 Output elasticities We can obtain output elasticities based on the quadratic production function by the standard formula for output elasticities: i = M Pi

xi y

(2.131)

As explained in section 2.4.11.1, we will use the predicted output quantities rather than the observed output quantities. We can calculate the output elasticities with: > dat$eCapQuad dat$eLabQuad dat$eMatQuad hist( dat$eCapQuad, 15 ) > hist( dat$eLabQuad, 15 )

−0.4

0.0

0.4

0.8

50 40 30 0

0

0

10

20

Frequency

30 10

20

Frequency

30 20 10

Frequency

40

50

60

> hist( dat$eMatQuad, 15 )

−0.5 0.0

0.5

eCap

1.0

1.5

2.0

2.5

−1.5

eLab

−0.5

0.5 1.0 1.5 eMat

Figure 2.26: Quadratic production function: output elasticities The resulting graphs are shown in figure 2.26. If the firms increase capital input by one percent, the output of most firms will increase by around 0.05 percent. If the firms increase labor input by one percent, the output of most firms will increase by around 0.7 percent. If the firms increase material input by one percent, the output of most firms will increase by around 0.5 percent.

2.5.7 Elasticity of scale The elasticity of scale can—as always—be calculated as the sum of all output elasticities. > dat$eScaleQuad hist( dat$eScaleQuad, 30 )

12 0.8

1.0

1.2

1.4

1.6

0 2 4 6 8

Frequency

25 15 0 5

Frequency

> hist( dat$eScaleQuad[ dat$monoQuad ], 30 )

1.1

eScaleQuad

1.3

1.5

1.7

eScaleQuad[ monoQuad ]

Figure 2.27: Quadratic production function: elasticities of scale The resulting graphs are shown in figure 2.27. Only a very few firms (4 out of 140) experience decreasing returns to scale. If we only consider the observations where all monotonicity conditions are fulfilled, our results suggest that all firms have increasing returns to scale. Most firms have an elasticity of scale around 1.3. Hence, if these firms increase all input quantities by one percent, the output of most firms will increase by around 1.3 percent. These elasticities of scale are much more realistic than the elasticities of scale based on the linear production function. Information on the optimal firm size can be obtained by analyzing the interrelationship between firm size and the elasticity of scale, where we can either use the observed output or the quantity index of the inputs as proxies of the firm size: > plot( dat$qOut, dat$eScaleQuad, log = "x" ) > abline( 1, 0 ) > plot( dat$X, dat$eScaleQuad, log = "x" ) > abline( 1, 0 ) > plot( dat$qOut[ dat$monoQuad ], dat$eScaleQuad[ dat$monoQuad ], log = "x" ) > plot( dat$X[ dat$monoQuad ], dat$eScaleQuad[ dat$monoQuad ], log = "x" ) The resulting graphs are shown in figure 2.28. They all indicate that there are increasing returns to scale for all firm sizes in the sample. Hence, all firms in the sample would gain from increasing their size and the optimal firm size seems to be larger than the largest firm in the sample.

2.5.8 Marginal rates of technical substitution We can calculate the marginal rates of technical substitution (MRTS) based on our estimated quadratic production function by following commands:

105

1e+05

5e+05

2e+06

0.8 1.0 1.2 1.4 1.6

eScaleQuad

0.8 1.0 1.2 1.4 1.6

eScaleQuad

2 Primal Approach: Production Function

1e+07

0.5

5e+05

2e+06

5.0

1.7 1.5 1.3 1.1

eScaleQuad[ monoQuad ]

1.7 1.5 1.3 1e+05

2.0

quantity index of inputs

1.1

eScaleQuad[ monoQuad ]

observed output

1.0

5e+06

0.5

observed output

1.0

2.0

quantity index of inputs

Figure 2.28: Quadratic production function: elasticities of scale at different firm sizes > dat$mrtsCapLabQuad dat$mrtsLabCapQuad dat$mrtsCapMatQuad dat$mrtsMatCapQuad dat$mrtsLabMatQuad dat$mrtsMatLabQuad hist( dat$mrtsCapLabQuad[ dat$monoQuad ], 30 ) > hist( dat$mrtsLabCapQuad[ dat$monoQuad ], 30 ) > hist( dat$mrtsCapMatQuad[ dat$monoQuad ], 30 ) > hist( dat$mrtsMatCapQuad[ dat$monoQuad ], 30 ) > hist( dat$mrtsLabMatQuad[ dat$monoQuad ], 30 ) > hist( dat$mrtsMatLabQuad[ dat$monoQuad ], 30 ) The resulting graphs are shown in figure 2.29. As some outliers hide the variation of the majority of the RMRTS, we use function colMedians (package miscTools) to show the median values of the MRTS:

106

−40

−20

0

80 60 20 0

10 0

0 −60

40

Frequency

30 20

Frequency

30 20 10

Frequency

40

40

50

2 Primal Approach: Production Function

−15

−5

0

−1000

−600

mrtsLabCapQuad

−200

0

mrtsCapMatQuad

−5

−4

−3

−2

−1

0

10 0

5 0 −6

20

Frequency

10

Frequency

40 0

20

Frequency

60

30

15

80

mrtsCapLabQuad

−10

−60

mrtsMatCapQuad

−40

−20

0

−3

mrtsLabMatQuad

−2

−1

0

mrtsMatLabQuad

Figure 2.29: Quadratic production function: marginal rates of technical substitution (RMRTS) > colMedians( subset( dat, monoQuad, + +

c( "mrtsCapLabQuad", "mrtsLabCapQuad", "mrtsCapMatQuad", "mrtsMatCapQuad", "mrtsLabMatQuad", "mrtsMatLabQuad" ) ) )

mrtsCapLabQuad mrtsLabCapQuad mrtsCapMatQuad mrtsMatCapQuad mrtsLabMatQuad -2.23505371

-0.44741654

-14.19802214

-0.07043235

-7.86423950

mrtsMatLabQuad -0.12715788 Given that the median marginal rate of technical substitution between capital and labor is -2.24, a typical firm that reduces the use of labor by one unit, has to use around 2.24 additional units of capital in order to produce the same amount of output as before. Alternatively, the typical firm can replace one unit of labor by using 0.13 additional units of materials.

2.5.9 Relative marginal rates of technical substitution As we do not have a practical interpretation of the units of measurement of the input quantities, the relative marginal rates of technical substitution (RMRTS) are practically more meaningful than the MRTS. The following commands calculate the RMRTS:

107

2 Primal Approach: Production Function > dat$rmrtsCapLabQuad dat$rmrtsLabCapQuad dat$rmrtsCapMatQuad dat$rmrtsMatCapQuad dat$rmrtsLabMatQuad dat$rmrtsMatLabQuad hist( dat$rmrtsCapLabQuad[ dat$monoQuad ], 30 ) > hist( dat$rmrtsLabCapQuad[ dat$monoQuad ], 30 ) > hist( dat$rmrtsCapMatQuad[ dat$monoQuad ], 30 ) > hist( dat$rmrtsMatCapQuad[ dat$monoQuad ], 30 ) > hist( dat$rmrtsLabMatQuad[ dat$monoQuad ], 30 )

0

−15

−10

−5

0

−700

rmrtsLabCapQuad

Frequency

15

−100

rmrtsCapMatQuad

5

10

10

Frequency

−300

25

80 60 40 −40

−30

−20

−10

rmrtsMatCapQuad

0

0

0

0

5

20

Frequency

−500

30

20

rmrtsCapLabQuad

60 20

−20

35

−200

20

−400

15

−600

0

0 −800

40

Frequency

20

40

Frequency

40 0

20

Frequency

60

60

80

80

80

> hist( dat$rmrtsMatLabQuad[ dat$monoQuad ], 30 )

−6

−4

−2

rmrtsLabMatQuad

0

−15

−10

−5

0

rmrtsMatLabQuad

Figure 2.30: Quadratic production function: relative marginal rates of technical substitution (RMRTS) The resulting graphs are shown in figure 2.30. As some outliers hide the variation of the majority of the RMRTS, we use function colMedians (package miscTools) to show the median values of

108

2 Primal Approach: Production Function the RMRTS: > colMedians( subset( dat, monoQuad, +

c( "rmrtsCapLabQuad", "rmrtsLabCapQuad", "rmrtsCapMatQuad",

+

"rmrtsMatCapQuad", "rmrtsLabMatQuad", "rmrtsMatLabQuad" ) ) )

rmrtsCapLabQuad rmrtsLabCapQuad rmrtsCapMatQuad rmrtsMatCapQuad rmrtsLabMatQuad -5.5741780

-0.1793986

-4.2567577

-0.2349206

-0.7745132

rmrtsMatLabQuad -1.2911336 Given that the median relative marginal rate of technical substitution between capital and labor is -5.57, a typical firm that reduces the use of labor by one percent, has to use around 5.57 percent more capital in order to produce the same amount of output as before. Alternatively, the typical firm can replace one percent of labor by using 1.29 percent more materials.

2.5.10 Quasiconcavity In order to check whether the estimated quadratic production function is quasiconcave, we need to construct the bordered Hessian matrix. As the first derivatives of the quadratic production function differ between observations, also the bordered Hessian matrix differs between observations. As a starting point, we construct the bordered Hessian matrix just for the first observation: > bhm bhm[ 1, 2 ] range( with( dat, qCap * mpCapQuad * esaCapLabQuad + +

qLab * mpLabQuad * esaLabLabQuad + qMat * mpMatQuad * esaLabMatQuad ) )

[1] -3.725290e-09

2.095476e-09

> range( with( dat, qCap * mpCapQuad * esaCapMatQuad + +

qLab * mpLabQuad * esaLabMatQuad + qMat * mpMatQuad * esaMatMatQuad ) )

[1] -1.117587e-08

1.396984e-09

The extremely small deviations from zero are most likely caused by rounding errors that are unavoidable on digital computers. This test does not prove that all our calculations are done correctly but if we had made a mistake, we would have discovered it with a very high probability. Hence, we can be rather sure that our calculations are correct. As for the direct elasticities of substitution, we visualize (the variation of) the Allen elasticities of substitution only for the observations, where the monotonicity condition is fulfilled: > hist( dat$esaCapLabQuad[ dat$monoQuad ], 30 ) > hist( dat$esaCapMatQuad[ dat$monoQuad ], 30 ) > hist( dat$esaLabMatQuad[ dat$monoQuad ], 30 ) The resulting graphs are shown in figure 2.32. As the Allen elasticities of substitution allow the other inputs to adjust, a meaningful interpretation requires that not only the partial isoquants of the two considered inputs are convex but that the production is quasiconcave. As the estimated quadratic production is not quasiconcave at any observation, we do not interpret the obtained Allen elasticities of substitution.

114

25 20 15 10

Frequency

15 5

10

Frequency

30 20

−8

−6

−4

−2

0

0

0

0

5

10

Frequency

40

50

20

2 Primal Approach: Production Function

−2

esaCapLabQuad

−1

0

1

2

3

4

0.0

0.5

esaCapMatQuad

1.0

1.5

2.0

2.5

esaLabMatQuad

Figure 2.32: Quadratic production function: Allen elasticities of substitution 2.5.11.3 Comparison of direct and Allen elasticities of substitution In the following, we use scatter plots to compare the estimated direct elasticities of substitution with the estimated Allen elasticities of substitution: > compPlot( dat$esdCapLabQuad[ dat$monoQuad ], +

dat$esaCapLabQuad[ dat$monoQuad ], lim = c( -4, 1 ) )

> compPlot( dat$esdCapMatQuad[ dat$monoQuad ], +

dat$esaCapMatQuad[ dat$monoQuad ], lim = c( -10, 5) )

> compPlot( dat$esdLabMatQuad[ dat$monoQuad ],

−4

−3

−2

−1

esdCapLabQuad

0

1

2.0 1.5 0.0

−10

0.5

1.0

esaLabMatQuad

0 −5

esaCapMatQuad

−1 −2 −3 −4

esaCapLabQuad

0

2.5

5

dat$esaLabMatQuad[ dat$monoQuad ] ) 1

+

−10

−5

0

esdCapMatQuad

5

0.0

0.5

1.0

1.5

2.0

2.5

esdLabMatQuad

Figure 2.33: Quadratic production function: Comparison of direct and Allen elasticities of substitution The resulting graphs are shown in figure 2.33. We focus on the elasticities of substitution between labor and materials, because the elasticities of substitution between capital and labor and the elasticities of substitution between capital and material cannot be meaningful interpreted (see section 2.5.11.1). In contrast to the direct elasticities of substitution, the Allen elasticities of substitution allow the other inputs to adjust. Therefore, it is not surprising that the Allen

115

2 Primal Approach: Production Function elasticities of substitution between labor and materials are about the same or a little larger than the direct elasticities of substitution between labor and materials. However, even the Allen elasticities of substitution between labor and materials indicate a low substitutability between labor and materials (see also section 2.5.11.1).

2.5.12 First-order conditions for profit maximization In this section, we will check to what extent the first-order conditions for profit maximization (2.40) are fulfilled, i.e. to what extent the firms use the optimal input quantities. We do this by comparing the marginal value products of the inputs with the corresponding input prices. We can calculate the marginal value products by multiplying the marginal products by the output price: > dat$mvpCapQuad dat$mvpLabQuad dat$mvpMatQuad compPlot( dat$pCap, dat$mvpCapQuad ) > compPlot( dat$pLab, dat$mvpLabQuad ) > compPlot( dat$pMat, dat$mvpMatQuad ) > compPlot( dat$pCap[ dat$monoQuad ], dat$mvpCapQuad[ dat$monoQuad ], log = "xy" ) > compPlot( dat$pLab[ dat$monoQuad ], dat$mvpLabQuad[ dat$monoQuad ], log = "xy" ) > compPlot( dat$pMat[ dat$monoQuad ], dat$mvpMatQuad[ dat$monoQuad ], log = "xy" ) The resulting graphs are shown in figure 2.34. They indicate that the marginal value products of most firms are higher than the corresponding input prices. This indicates that most firms could increase their profit by using more of all inputs. Given that the estimated quadratic function shows that (almost) all firms operate under increasing returns to scale, it is not surprising that most firms would gain from increasing all input quantities. Therefore, the question arises why the firms in the sample did not do this. This questions has already been addressed in section 2.3.10.

2.5.13 First-order conditions for cost minimization As the marginal rates of technical substitution differ between observations for the three other functional forms, we use scatter plots for visualizing the comparison of the input price ratios with the negative inverse marginal rates of technical substitution. As the marginal rates of technical substitution are meaningless if the monotonicity condition is not fulfilled, we limit the comparisons to the observations, where all monotonicity conditions are fulfilled:

116

−60

−40

−20

0

300 0

100

200

MVP Mat

0

−60

10

20

MVP Lab

−20 −40

MVP Cap

30

400

0

500

40

2 Primal Approach: Production Function

0

10

30

40

0

100

300

w Lab

500

w Mat

50 20 10

0.5

5

1.0

2.0

MVP Mat

5.0 10.0

MVP Lab

2.0 0.5 0.1

MVP Cap

5.0

100

w Cap

20

0.1

0.5

2.0

5.0

0.5 1.0 2.0

w Cap

5.0

20.0

5

10

w Lab

Figure 2.34: Marginal value products and corresponding input prices

117

20 w Mat

50

100

2 Primal Approach: Production Function > compPlot( ( dat$pCap / dat$pLab )[ dat$monoQuad ], +

- dat$mrtsLabCapQuad[ dat$monoQuad ] )

> compPlot( ( dat$pCap / dat$pMat )[ dat$monoQuad ], +

- dat$mrtsMatCapQuad[ dat$monoQuad ] )

> compPlot( ( dat$pLab / dat$pMat )[ dat$monoQuad ], +

- dat$mrtsMatLabQuad[ dat$monoQuad ] )

> compPlot( ( dat$pCap / dat$pLab )[ dat$monoQuad ], +

- dat$mrtsLabCapQuad[ dat$monoQuad ], log = "xy" )

> compPlot( ( dat$pCap / dat$pMat )[ dat$monoQuad ], +

- dat$mrtsMatCapQuad[ dat$monoQuad ], log = "xy" )

> compPlot( ( dat$pLab / dat$pMat )[ dat$monoQuad ], - dat$mrtsMatLabQuad[ dat$monoQuad ], log = "xy" )

5

10

2 1

− MRTS Mat Lab

4 3

15

0

1

4

5

0.10

0.50 2.00

0

10.00

1

2

3

2.00

w Lab / w Mat

0.02 0.05

0.001 0.02

6

− MRTS Mat Lab

1.000 0.100 0.010

− MRTS Mat Cap

2.00 0.50 0.10 0.02

− MRTS Lab Cap

3

w Cap / w Mat

10.00

w Cap / w Lab

2

0.20 0.50

0

0

0

0

1

2

− MRTS Mat Cap

10 5

− MRTS Lab Cap

5

3

6

15

+

0.001

0.010

w Cap / w Lab

0.100

1.000

0.02

w Cap / w Mat

0.10

0.50

2.00

w Lab / w Mat

Figure 2.35: First-order conditions for costs minimization The resulting graphs are shown in figure 2.35. Furthermore, we use histograms to visualize the (absolute and relative) differences between the input price ratios and the corresponding negative inverse marginal rates of technical substitution: > hist( ( - dat$mrtsLabCapQuad - dat$pCap / dat$pLab )[ dat$monoQuad ] )

118

2 Primal Approach: Production Function > hist( ( - dat$mrtsMatCapQuad - dat$pCap / dat$pMat )[ dat$monoQuad ] ) > hist( ( - dat$mrtsMatLabQuad - dat$pLab / dat$pMat )[ dat$monoQuad ] ) > hist( log( - dat$mrtsLabCapQuad / ( dat$pCap / dat$pLab ) )[ dat$monoQuad ] ) > hist( log( - dat$mrtsMatCapQuad / (dat$pCap / dat$pMat ) )[ dat$monoQuad ] )

40 20 10

20 0

5

10

0

0

10 0 −5

15

0

2

4

6

0

− MrtsMatCap − wCap / wMat

2

3

4

−6

−4

−2

0

2

4

log(−MrtsLabCap / (wCap / wLab))

40 10

20

Frequency

15 10

0

0

0

5

5

10

15

Frequency

20

30

20

25

1

− MrtsMatLab − wLab / wMat

25

30

− MrtsLabCap − wCap / wLab

Frequency

30

Frequency

40

Frequency

40 30 20

Frequency

50

50

60

60

60

70

> hist( log( - dat$mrtsMatLabQuad / (dat$pLab / dat$pMat ) )[ dat$monoQuad ] )

−6

−4

−2

0

2

4

6

log(−MrtsMatCap / (wCap / wMat))

−3 −2 −1

0

1

2

3

4

log(−MrtsMatLab / (Lab / wMat))

Figure 2.36: First-order conditions for costs minimization The resulting graphs are shown in figure 2.36. The left graphs in figures 2.35 and 2.36 show that the ratio between the capital price and the labor price is larger than the absolute value of the marginal rate of technical substitution between labor and capital for a majority of the firms in the sample: wcap M Pcap > −M RT Slab,cap = wlab M Plab

(2.133)

Hence, these firms can get closer to the minimum of their production costs by substituting labor for capital, because this will decrease the marginal product of labor and increase the marginal product of capital so that the absolute value of the MRTS between labor and capital increases and gets closer to the corresponding input price ratio. Similarly, the graphs in the middle column indicate that a majority of the firms should substitute materials for capital and the graphs on

119

2 Primal Approach: Production Function the right indicate that a little more than half of the firms should substitute materials for labor. Hence, the majority of the firms could reduce production costs particularly by using less capital and using more labor or more materials.

6

2.6 Translog production function 2.6.1 Specification The Translog function is a more flexible extension of the Cobb-Douglas function as the quadratic function is a more flexible extension of the linear function. Hence, the Translog function can be seen as a combination of the Cobb-Douglas function and the quadratic function. The Translog production function has following specification: ln y = α0 +

X

1 XX αij ln xi ln xj 2 i j

αi ln xi +

i

with αij = αji .

2.6.2 Estimation We can estimate this Translog production function with the command > prodTL summary( prodTL ) Call: lm(formula = log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qLab)) + I(log(qCap) * log(qMat)) + I(log(qLab) * log(qMat)), data = dat) Residuals: Min

1Q

Median

3Q

Max

-1.68015 -0.36688

0.05389

0.44125

1.26560

Coefficients: Estimate Std. Error t value Pr(>|t|) 6

This generally confirms the results of the Cobb-Douglas production function.

120

(2.134)

2 Primal Approach: Production Function (Intercept)

-4.14581

21.35945

-0.194

0.8464

log(qCap)

-2.30683

2.28829

-1.008

0.3153

log(qLab)

1.99328

4.56624

0.437

0.6632

log(qMat)

2.23170

3.76334

0.593

0.5542

I(0.5 * log(qCap)^2)

-0.02573

0.20834

-0.124

0.9019

I(0.5 * log(qLab)^2)

-1.16364

0.67943

-1.713

0.0892 .

I(0.5 * log(qMat)^2)

-0.50368

0.43498

-1.158

0.2490

0.56194

0.29120

1.930

0.0558 .

I(log(qCap) * log(qMat)) -0.40996

0.23534

-1.742

0.0839 .

I(log(qLab) * log(qMat))

0.42750

1.539

I(log(qCap) * log(qLab))

0.65793

0.1262

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.6412 on 130 degrees of freedom Multiple R-squared:

0.6296,

F-statistic: 24.55 on 9 and 130 DF,

Adjusted R-squared:

0.6039

p-value: < 2.2e-16

None of the estimated coefficients is statistically significantly different from zero at the 5% significance level and only three coefficients are statistically significant at the 10% level. As the Cobb-Douglas production function is “nested” in the Translog production function, we can apply a “Wald test” or “likelihood ratio test” to check whether the Cobb-Douglas production function is rejected in favor of the Translog production function. This can be done by the functions waldtest and lrtest (package lmtest): > waldtest( prodCD, prodTL ) Wald test Model 1: log(qOut) ~ log(qCap) + log(qLab) + log(qMat) Model 2: log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qLab)) + I(log(qCap) * log(qMat)) + I(log(qLab) * log(qMat)) Res.Df Df 1

136

2

130

F

Pr(>F)

6 2.062 0.06202 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> lrtest( prodCD, prodTL ) Likelihood ratio test

121

2 Primal Approach: Production Function Model 1: log(qOut) ~ log(qCap) + log(qLab) + log(qMat) Model 2: log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qLab)) + I(log(qCap) * log(qMat)) + I(log(qLab) * log(qMat)) #Df

LogLik Df

1

5 -137.61

2

11 -131.25

Chisq Pr(>Chisq)

6 12.727

0.04757 *

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

At the 5% significance level, the Cobb-Douglas production function is accepted by the Wald test but rejected in favor of the Translog production function by the likelihood ratio test. In order to reduce the chance of using a too restrictive functional form, we proceed with the Translog production function.

2.6.3 Statistical significance of individual inputs The following code applies Wald tests (using R functions linearHypothesis and waldtest) and Likelihood Ratio tests (using the R function lrtest) to test whether each of the three input quantities significanctly affects the output quantity in the estimated Translog production function. > library( "car" ) > # capital > linearHypothesis( prodTL, c( "log(qCap) = 0", "I(0.5 * log(qCap)^2) = 0", +

"I(log(qCap) * log(qLab)) = 0", "I(log(qCap) * log(qMat)) = 0" ) )

Linear hypothesis test Hypothesis: log(qCap) = 0 I(0.5 * log(qCap)^2) = 0 I(log(qCap) * log(qLab)) = 0 I(log(qCap) * log(qMat)) = 0 Model 1: restricted model Model 2: log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qLab)) + I(log(qCap) * log(qMat)) + I(log(qLab) * log(qMat)) Res.Df

RSS Df Sum of Sq

F Pr(>F)

122

2 Primal Approach: Production Function 1

134 57.039

2

130 53.447

4

3.5916 2.184 0.0743 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> waldtest( prodTL, log( qOut ) ~ log( qLab ) + log( qMat ) +

+ I( 0.5 * log( qLab )^2 ) + I( 0.5 * log( qMat )^2 )

+

+ I( log( qLab ) * log( qMat ) ) )

Wald test Model 1: log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qLab)) + I(log(qCap) * log(qMat)) + I(log(qLab) * log(qMat)) Model 2: log(qOut) ~ log(qLab) + log(qMat) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qLab) * log(qMat)) Res.Df Df

F Pr(>F)

1

130

2

134 -4 2.184 0.0743 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> lrtest( prodTL, log( qOut ) ~ log( qLab ) + log( qMat ) +

+ I( 0.5 * log( qLab )^2 ) + I( 0.5 * log( qMat )^2 )

+ +

+ I( log( qLab ) * log( qMat ) ) )

Likelihood ratio test Model 1: log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qLab)) + I(log(qCap) * log(qMat)) + I(log(qLab) * log(qMat)) Model 2: log(qOut) ~ log(qLab) + log(qMat) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qLab) * log(qMat)) #Df 1

LogLik Df

Chisq Pr(>Chisq)

11 -131.25

2

7 -135.80 -4 9.1052

0.05852 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> # labor > linearHypothesis( prodTL, c( "log(qLab) = 0", "I(0.5 * log(qLab)^2) = 0",

123

2 Primal Approach: Production Function +

"I(log(qCap) * log(qLab)) = 0", "I(log(qLab) * log(qMat)) = 0" ) )

Linear hypothesis test Hypothesis: log(qLab) = 0 I(0.5 * log(qLab)^2) = 0 I(log(qCap) * log(qLab)) = 0 I(log(qLab) * log(qMat)) = 0 Model 1: restricted model Model 2: log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qLab)) + I(log(qCap) * log(qMat)) + I(log(qLab) * log(qMat)) Res.Df

RSS Df Sum of Sq

1

134 66.649

2

130 53.447

4

F

Pr(>F)

13.202 8.028 8.14e-06 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> waldtest( prodTL, log( qOut ) ~ log( qCap ) + log( qMat ) +

+ I( 0.5 * log( qCap )^2 ) + I( 0.5 * log( qMat )^2 )

+

+ I( log( qCap ) * log( qMat ) ) )

Wald test Model 1: log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qLab)) + I(log(qCap) * log(qMat)) + I(log(qLab) * log(qMat)) Model 2: log(qOut) ~ log(qCap) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qMat)) Res.Df Df

F

Pr(>F)

1

130

2

134 -4 8.028 8.14e-06 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> lrtest( prodTL, log( qOut ) ~ log( qCap ) + log( qMat ) +

+ I( 0.5 * log( qCap )^2 ) + I( 0.5 * log( qMat )^2 )

+

+ I( log( qCap ) * log( qMat ) ) )

124

2 Primal Approach: Production Function Likelihood ratio test Model 1: log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qLab)) + I(log(qCap) * log(qMat)) + I(log(qLab) * log(qMat)) Model 2: log(qOut) ~ log(qCap) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qMat)) #Df 1

LogLik Df

Chisq Pr(>Chisq)

11 -131.25

2

7 -146.70 -4 30.905

3.201e-06 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> # materials > linearHypothesis( prodTL, c( "log(qMat) = 0", "I(0.5 * log(qMat)^2) = 0", +

"I(log(qCap) * log(qMat)) = 0", "I(log(qLab) * log(qMat)) = 0" ) )

Linear hypothesis test Hypothesis: log(qMat) = 0 I(0.5 * log(qMat)^2) = 0 I(log(qCap) * log(qMat)) = 0 I(log(qLab) * log(qMat)) = 0 Model 1: restricted model Model 2: log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qLab)) + I(log(qCap) * log(qMat)) + I(log(qLab) * log(qMat)) Res.Df

RSS Df Sum of Sq

1

134 68.178

2

130 53.447

4

F

Pr(>F)

14.731 8.9578 2.02e-06 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> waldtest( prodTL, log( qOut ) ~ log( qCap ) + log( qLab ) +

+ I( 0.5 * log( qCap )^2 ) + I( 0.5 * log( qLab )^2 )

+

+ I( log( qCap ) * log( qLab ) ) )

Wald test

125

2 Primal Approach: Production Function Model 1: log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qLab)) + I(log(qCap) * log(qMat)) + I(log(qLab) * log(qMat)) Model 2: log(qOut) ~ log(qCap) + log(qLab) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(log(qCap) * log(qLab)) Res.Df Df

F

Pr(>F)

1

130

2

134 -4 8.9578 2.02e-06 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> lrtest( prodTL, log( qOut ) ~ log( qCap ) + log( qLab ) +

+ I( 0.5 * log( qCap )^2 ) + I( 0.5 * log( qLab )^2 )

+

+ I( log( qCap ) * log( qLab ) ) )

Likelihood ratio test Model 1: log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(0.5 * log(qMat)^2) + I(log(qCap) * log(qLab)) + I(log(qCap) * log(qMat)) + I(log(qLab) * log(qMat)) Model 2: log(qOut) ~ log(qCap) + log(qLab) + I(0.5 * log(qCap)^2) + I(0.5 * log(qLab)^2) + I(log(qCap) * log(qLab)) #Df 1

LogLik Df

Chisq Pr(>Chisq)

11 -131.25

2

7 -148.28 -4 34.081

7.173e-07 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

All inputs have a statistically significant effect at the 10% significance level, while labour and materials also have statistically significant effects at the 5% and 1% (and at even higher) significance levels.

2.6.4 Properties We cannot see from the estimated coefficients whether the monotonicity condition is fulfilled. The Translog production function cannot be globally monotone, because there will be always a set of input quantities that result in negative marginal products.7 The Translog function would only be globally monotone, if all first-order coefficients are positive and all second-order coefficients are zero, which is equivalent to a Cobb-Douglas function. We will check the monotonicity condition at each observation in section 2.6.6. 7

Please note that ln xj is a large negative number if xj is a very small positive number.

126

2 Primal Approach: Production Function All Translog production functions fulfill the weak and the strong essentiality assumption, because as soon as a single input quantity approaches zero, the right-hand side of equation (2.134) approaches minus infinity (if monotonicity is fulfilled), and thus, the output quantity y = exp(ln y) approaches zero. Hence, if a data set includes observations with a positive output quantity but at least one input quantity that is zero, strict essentiality cannot be fulfilled in the underlying true production technology so that the Translog production function is not a suitable functional form for analyzing this data set. The input requirement sets derived from Translog production functions are always closed and non-empty. The Translog production function always returns finite, real, non-negative, and single values as long as all input quantities are strictly positive. All Translog production functions are continuous and twice-continuously differentiable.

2.6.5 Predicted output quantities As before, we can easily obtain the predicted output quantities with the fitted method. As we used the logarithmic output quantity as dependent variable in our estimated model, we must use the exponential function to obtain the output quantities measured in levels: > dat$qOutTL compPlot( dat$qOut, dat$qOutTL )

0.0e+00

1e+07 1e+05

5e+05 2e+06

fitted

1.0e+07 0.0e+00

fitted

2.0e+07

> compPlot( dat$qOut, dat$qOutTL, log = "xy" )

1.0e+07

2.0e+07

1e+05

observed

5e+05

5e+06

observed

Figure 2.37: Translog production function: fit of the model The resulting graphs are shown in figure 2.37. While the graph in the left panel uses a linear scale for the axes, the graph in the right panel uses a logarithmic scale for both axes. Hence,

127

2 Primal Approach: Production Function the deviations from the 45°-line illustrate the absolute deviations in the left panel and the relative deviations in the right panel. The fit of the model looks rather okay, but there are some observations, at which the predicted output quantity is not very close to the observed output quantity.

2.6.6 Output elasticities The output elasticities calculated from a Translog production function are: i =

X ∂ ln y = αi + αij ln xj ∂ ln xi j

(2.135)

We can simplify the code for computing these output elasticities by using short names for the coefficients: > a1 a2 a3 a11 a22 a33 a12 table( dat$eCapTL >= 0 ) FALSE

TRUE

32

108

> table( dat$eLabTL >= 0 ) FALSE

TRUE

14

126

> table( dat$eMatTL >= 0 ) FALSE

TRUE

8

132

> dat$monoTL = 0 & eLabTL >= 0 & eMatTL >= 0 ) > table( dat$monoTL ) FALSE

TRUE

48

92

32 firms have a negative output elasticity of capital, 14 firms have a negative output elasticity of labor, and 8 firms have a negative output elasticity of materials. In total the monotonicity condition is not fulfilled at 48 out of 140 observations. Although the monotonicity conditions are fulfilled for a large part of firms in our data set, these frequent violations indicate a possible model misspecification.

129

2 Primal Approach: Production Function

2.6.7 Marginal products The first derivatives (marginal products) of the Translog production function with respect to the input quantities are: 



X y ∂ ln y y  ∂y = = αi + αij ln xj  M Pi = ∂xi xi ∂ ln xi xi j

(2.136)

We can calculate the marginal products based on the output elasticities that we have calculated above. As argued in section 2.4.11.1, we use the predicted output quantities in this calculation: > dat$mpCapTL dat$mpLabTL dat$mpMatTL hist( dat$mpCapTL, 15 ) > hist( dat$mpLabTL, 15 )

−10

0

10

20

mpCapTL

15 0

5

10

Frequency

20

20 15 0

0

5

10

Frequency

15 10 5

Frequency

20

25

> hist( dat$mpMatTL, 15 )

−5

0

5

10

15

20

25

0

mpLabTL

50

100

mpMatTL

Figure 2.39: Translog production function: marginal products The resulting graphs are shown in figure 2.39. If the firms increase capital input by one unit, the output of most firms will increase by around 4 units. If the firms increase labor input by one unit, the output of most firms will increase by around 4 units. If the firms increase material input by one unit, the output of most firms will increase by around 70 units.

2.6.8 Elasticity of scale The elasticity of scale can—as always—be calculated as the sum of all output elasticities. > dat$eScaleTL hist( dat$eScaleTL, 30 )

8 6 0

0

2

4

Frequency

10 5

Frequency

15

> hist( dat$eScaleTL[ dat$monoTL ], 30 )

1.2 1.3 1.4 1.5 1.6 1.7

1.2

eScaleTL

1.3

1.4

1.5

1.6

1.7

eScaleTL[ monoTL ]

Figure 2.40: Translog production function: elasticities of scale The resulting graphs are shown in figure 2.40. All firms experience increasing returns to scale and most of them have an elasticity of scale around 1.45. Hence, if these firms increase all input quantities by one percent, the output of most firms will increase by around 1.45 percent. These elasticities of scale are realistic and on average close to the elasticity of scale obtained from the Cobb-Douglas production function (1.47). Information on the optimal firm size can be obtained by analyzing the relationship between firm size and the elasticity of scale. We can either use the observed or the predicted output: > plot( dat$qOut, dat$eScaleTL, log = "x" ) > plot( dat$X, dat$eScaleTL, log = "x" ) > plot( dat$qOut[ dat$monoTL ], dat$eScaleTL[ dat$monoTL ], log = "x" ) > plot( dat$X[ dat$monoTL ], dat$eScaleTL[ dat$monoTL ], log = "x" ) The resulting graphs are shown in figure 2.41. Both of them indicate that the elasticity of scale slightly decreases with firm size but there are considerable increasing returns to scale even for the largest firms in the sample. Hence, all firms in the sample would gain from increasing their size and the optimal firm size seems to be larger than the largest firm in the sample.

2.6.9 Marginal rates of technical substitution We can calculate the marginal rates of technical substitution (MRTS) based on our estimated Translog production function by following commands: > dat$mrtsCapLabTL dat$mrtsLabCapTL dat$mrtsCapMatTL dat$mrtsMatCapTL dat$mrtsLabMatTL dat$mrtsMatLabTL hist( dat$mrtsCapLabTL[ dat$monoTL ], 30 ) > hist( dat$mrtsLabCapTL[ dat$monoTL ], 30 ) > hist( dat$mrtsCapMatTL[ dat$monoTL ], 30 ) > hist( dat$mrtsMatCapTL[ dat$monoTL ], 30 ) > hist( dat$mrtsLabMatTL[ dat$monoTL ], 30 ) > hist( dat$mrtsMatLabTL[ dat$monoTL ], 30 ) The resulting graphs are shown in figure 2.43. As some outliers hide the variation of the majority of the MRTS, we use function colMedians (package miscTools) to show the median values of the MRTS: > colMedians( subset( dat, monoTL,

132

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10 20 30 40 50 60 70

Frequency

40 30 0

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2 Primal Approach: Production Function

0

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0

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mrtsLabCapTL

−600

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mrtsCapMatTL

−0.6

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30 0

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20

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60 0

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mrtsCapLabTL

−40

−1200

mrtsMatCapTL

−800

−400

0

−4

mrtsLabMatTL

−3

−2

−1

0

mrtsMatLabTL

Figure 2.42: Translog production function: marginal rates of technical substitution (MRTS) + +

c( "mrtsCapLabTL", "mrtsLabCapTL", "mrtsCapMatTL", "mrtsMatCapTL", "mrtsLabMatTL", "mrtsMatLabTL" ) ) )

mrtsCapLabTL mrtsLabCapTL mrtsCapMatTL mrtsMatCapTL mrtsLabMatTL mrtsMatLabTL -0.83929283

-1.19196521 -12.72554396

-0.07858435 -12.79850828

-0.07813810

Given that the median marginal rate of technical substitution between capital and labor is -0.84, a typical firm that reduces the use of labor by one unit, has to use around 0.84 additional units of capital in order to produce the same amount of output as before. Alternatively, the typical firm can replace one unit of labor by using 0.08 additional units of materials.

2.6.10 Relative marginal rates of technical substitution As we do not have a practical interpretation of the units of measurement of the input quantities, the relative marginal rates of technical substitution (RMRTS) are practically more meaningful than the MRTS. The following commands calculate the RMRTS: > dat$rmrtsCapLabTL dat$rmrtsLabCapTL dat$rmrtsCapMatTL dat$rmrtsMatCapTL dat$rmrtsLabMatTL dat$rmrtsMatLabTL hist( dat$rmrtsCapLabTL[ dat$monoTL ], 30 ) > hist( dat$rmrtsLabCapTL[ dat$monoTL ], 30 ) > hist( dat$rmrtsCapMatTL[ dat$monoTL ], 30 ) > hist( dat$rmrtsMatCapTL[ dat$monoTL ], 30 ) > hist( dat$rmrtsLabMatTL[ dat$monoTL ], 30 )

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rmrtsLabCapTL

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rmrtsCapMatTL

40 30 20

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40 30 20

−3.0

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0

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rmrtsCapLabTL

Frequency

60 20 0

0 −500

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Frequency

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Frequency

60 40 0

20

Frequency

60

80

80

> hist( dat$rmrtsMatLabTL[ dat$monoTL ], 30 )

−50 −40 −30 −20 −10

0

−35

rmrtsLabMatTL

−25

−15

−5

0

rmrtsMatLabTL

Figure 2.43: Translog production function: relative marginal rates of technical substitution (RMRTS) The resulting graphs are shown in figure 2.43. As some outliers hide the variation of the majority of the RMRTS, we use function colMedians (package miscTools) to show the median values of the RMRTS: > colMedians( subset( dat, monoTL, + +

c( "rmrtsCapLabTL", "rmrtsLabCapTL", "rmrtsCapMatTL", "rmrtsMatCapTL", "rmrtsLabMatTL", "rmrtsMatLabTL" ) ) )

134

2 Primal Approach: Production Function rmrtsCapLabTL rmrtsLabCapTL rmrtsCapMatTL rmrtsMatCapTL rmrtsLabMatTL -2.8357239

-0.3539150

-3.0064237

-0.3331325

-1.3444115

rmrtsMatLabTL -0.7439008 Given that the median relative marginal rate of technical substitution between capital and labor is -2.84, a typical firm that reduces the use of labor by one percent, has to use around 2.84 percent more capital in order to produce the same amount of output as before. Alternatively, the typical firm can replace one percent of labor by using 0.74 percent more materials.

2.6.11 Second partial derivatives In order to compute the elasticities of substitution, we need obtain the second derivatives of the Translog function. We can calculate them as derivatives of the first derivatives of the Translog function: ∂y ∂xi ∂xj



∂2y = ∂xi ∂xj

∂





∂ (αi + =

k αik ln xk )

y xi



(2.137)

∂xj

αi + αij y + = xj xi αij y αi + = + xi xj

P

P

X αik ln xk ∂y − δij αi + αik ln xk xi ∂xj k

k

P

αik ln xk xi

k

!

αj +

X k

αjk ln xk

!

y x2i

X y − δij αi + αik ln xk xj k

(2.138) !

y x2i

(2.139) αij y i j y i y = + − δij 2 xi xj xi xj xi y = (αij + i j − δij i ) , xi xj

(2.140) (2.141)

where δij is (again) Kronecker’s delta (2.95). Alternatively, the second derivatives of the Translog function can be expressed based on the marginal products (instead of the output elasticities): ∂2y αij y M Pi M Pj M Pi = + − δij ∂xi ∂xj xi xj y xi Now, we can calculate the second derivatives for each observation in our data set: > dat$fCapCapTL dat$fLabLabTL dat$fMatMatTL dat$fCapLabTL dat$fCapMatTL dat$fLabMatTL bhm bhm[ 1, 2 ] hist( dat$esdLabMatTL[ dat$monoTL & dat$quasiConcTL ], 30 )

1

2

esdCapLabTL

3

4

5

6

esdCapMatTL

0.00

0.10

0.20

0.30

esdLabMatTL

Figure 2.45: Translog production function: direct elasticities of substitution (at observations, where both the monotonicity condition and the quasiconcavity condition are fulfilled) The resulting graphs are shown in figure 2.45. The corresponding median values of the direct elasticities of substitution can be obtained by: > colMedians( subset( dat, dat$monoTL & dat$quasiConcTL, +

c( "esdCapLabTL", "esdCapMatTL", "esdLabMatTL" ) ) )

esdCapLabTL esdCapMatTL esdLabMatTL 0.2685804

1.9410943

0.2441392

For all firms in our sample, the estimated direct elasticities of substitution between capital and labor and between labor and materials are considerably smaller than the elasticity of substitution

139

2 Primal Approach: Production Function implied by the Cobb-Douglas production function (σ = 1) and are somewhat close to the elasticity of substitution implied by the Leontief production function (σ = 0). Hence, the substitutability between capital and labor as well as the substitutability between labor and materials seems to be rather low. The direct elasticities of substitution between capital and labor are for most firms around 0.27. Hence, if these firms keep the materials quantity and the output quantity unchanged and substitute capital for labor (or vice versa) so that the ratio between the capital quantity and the labor quantity increases (decreases) by around 0.27 percent, the MRTS between capital and labor increases (decreases) by one percent. If the ratio between the labor price and the capital price increases by one percent, firms that keep the materials input and the output quantity unchanged and minimize costs will substitute capital for labor so that the ratio between the capital quantity and the labor quantity increases by around 0.27 percent. Hence, the relative changes of the ratios of the input quantities are much smaller than the relative changes of the ratios of the corresponding input prices, which indicates a very low substitutability between capital and labor. Similarly, the direct elasticities of substitution between labor and materials are for most firms around 0.24. If these firms keep the capital quantity and the output quantity unchanged and substitute labor for materials (or vice versa) so that the ratio between the labor quantity and the quantity of materials increases (decreases) by around 0.24 percent, the MRTS between labor and materials increases (decreases) by one percent. If the ratio between the materials price and the labor price increases by one percent, firms that keep the capital input and the output quantity unchanged and minimize costs will substitute labor for materials so that the ratio between the labor quantity and the materials quantity increases by around 0.24 percent. Again, the relative changes of the ratios of the input quantities are much smaller than the relative changes of the ratios of the corresponding input prices, which indicates a very low substitutability between labor and materials. In contrast, the direct elasticities of substitution between capital and materials are for most firms around 1.94 and thus, considerably larger than the elasticity of substitution implied by the Cobb-Douglas production function (σ = 1). Hence, if these firms keep the labor quantity and the output quantity unchanged and substitute capital for materials (or vice versa) so that the ratio between the capital quantity and the quantity of materials increases (decreases) by around 1.94 percent, the MRTS between capital and materials increases (decreases) by one percent. If the ratio between the materials price and the capital price increases by one percent, firms that keep the labor input and the output quantity unchanged and minimize costs will substitute capital for materials so that the ratio between the capital quantity and the quantity of materials increases by around 1.94 percent. Hence, the relative change of the ratio of the input quantities is around twice as large as the relative change of the ratio of the input prices, which indicates a quite substantial substitutability between capital and materials.

140

2 Primal Approach: Production Function 2.6.13.2 Allen elasticities of substitution As for the quadratic production function, we only calculate the direct and Allen elasticities of substitution. The calculation of the Morishima elasticities of substitution requires only minimal changes of the code for calculating the Allen elasticities of substitution. In order to check whether our calculations are correct, we will—as before—check if the conditions (2.132) are fulfilled. In order to check these conditions, we need to calculate not only (normal) elasticities of substitution (σij ; i 6= j) but also economically not meaningful “elasticities of self-substitution” (σii ): > dat$FCapLabTL dat$FCapMatTL dat$FLabMatTL dat$FCapCapTL dat$FLabLabTL dat$FMatMatTL dat$bhmDetTL dat$numeratorTL dat$esaCapLabTL dat$esaCapMatTL dat$esaLabMatTL dat$esaCapCapTL dat$esaLabLabTL dat$esaMatMatTL range( with( dat, qCap * mpCapTL * esaCapCapTL + +

qLab * mpLabTL * esaCapLabTL + qMat * mpMatTL * esaCapMatTL ) )

[1] -5.960464e-08

1.907349e-06

> range( with( dat, qCap * mpCapTL * esaCapLabTL + +

qLab * mpLabTL * esaLabLabTL + qMat * mpMatTL * esaLabMatTL ) )

[1] -1.490116e-08

5.960464e-08

141

2 Primal Approach: Production Function > range( with( dat, qCap * mpCapTL * esaCapMatTL + +

qLab * mpLabTL * esaLabMatTL + qMat * mpMatTL * esaMatMatTL ) )

[1] -3.725290e-08

1.311302e-06

The extremely small deviations from zero are most likely caused by rounding errors that are unavoidable on digital computers. This test does not prove that all of our calculations are done correctly but if we had made a mistake, we probably would have discovered it. Hence, we can be rather sure that our calculations are correct. As explained above in section 2.6.13.1, the elasticities of substitution are meaningless if the monotonicity condition is violated and they are difficult to interpret if the quasiconcavity condition is violated. Hence, we visualize (the variation of) the Allen elasticities of substitution only for the observations, where both the monotonicity condition and the quasiconcavity condition are fulfilled: > hist( dat$esaCapLabTL[ dat$monoTL & dat$quasiConcTL ], 30 ) > hist( dat$esaCapMatTL[ dat$monoTL & dat$quasiConcTL ], 30 ) > hist( dat$esaLabMatTL[ dat$monoTL & dat$quasiConcTL ], 30 ) > hist( dat$esaCapLabTL[ dat$monoTL & dat$quasiConcTL & +

abs( dat$esaCapLabTL ) < 10 ], 30 )

> hist( dat$esaCapMatTL[ dat$monoTL & dat$quasiConcTL & +

abs( dat$esaCapMatTL ) < 10 ], 30 )

> hist( dat$esaLabMatTL[ dat$monoTL & dat$quasiConcTL & +

abs( dat$esaLabMatTL ) < 10 ], 30 )

The resulting graphs are shown in figure 2.46. The estimated elasticities of substitution between capital and labor suggest that capital and labor are complements for the majority of firms. In contrast, capital and materials are always substitutes and labor and materials are substitutes for the majority of firms. In order to avoid the effects of outliers, we use function colMedians (package miscTools) to obtain the median values of the Allen elasticities of substitution: > colMedians( subset( dat, monoTL & quasiConcTL, +

c( "esaCapLabTL", "esaCapMatTL", "esaLabMatTL" ) ) )

esaCapLabTL esaCapMatTL esaLabMatTL -0.8018194

3.6322399

0.5848447

The median elasticity of substitution between labor and materials (0.58) lies between the elasticity of substitution implied by the Leontief production function (σ = 0) and the elasticity of substitution implied by the Cobb-Douglas production function (σ = 1). Hence, the substitutability between labor and materials seems to be rather low. If a typical firm keeps the output quantity constant, substitutes materials for labor (or vice versa) so that the ratio between the quantity

142

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50 −300

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1000

1500

0

50

esaCapMatTL

100

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abs( esaCapLabTL ) < 10

0

1 0

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Frequency

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7

10 12 14

esaCapLabTL

Frequency

30 10 0

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Frequency

30 20 10

Frequency

40

2 Primal Approach: Production Function

2

3

4

5

6

7

8

abs( esaCapMatTL ) < 10

0

2

4

6

abs( esaLabMatTL ) < 10

Figure 2.46: Translog production function: Allen elasticities of substitution of materials and the labor quantity increases (decreases) by 0.58 percent, and adjusts the capital quantity accordingly, the MRTS between materials and labor will increase (decrease) by one percent. If the ratio between the labor price and the materials price increases by one percent, a firm that keeps the output quantity constant, adjusts all input quantities and minimizes costs, will substitute materials for labor so that the ratio between the quantity of materials and the labor quantity increases by 0.58 percent. Hence, the relative change of the ratio of the input quantities is considerably smaller than the relative change of the ratio of the input prices, which indicates a low substitutability between labor and materials. In contrast, the median elasticity of substitution between capital and materials is larger than one (3.63), which indicates that it is much easier to substitute between capital and materials. 2.6.13.3 Comparison of direct and Allen elasticities of substitution In the following, we use scatter plots to compare the estimated direct elasticities of substitution with the estimated Allen elasticities of substitution: > compPlot( dat$esdCapLabTL[ dat$monoTL & dat$quasiConcTL ], +

dat$esaCapLabTL[ dat$monoTL & dat$quasiConcTL ],

+

lim = c( -2, 2 ) )

> compPlot( dat$esdCapMatTL[ dat$monoTL & dat$quasiConcTL ],

143

2 Primal Approach: Production Function +

dat$esaCapMatTL[ dat$monoTL & dat$quasiConcTL ],

+

lim = c( -10, 10 ) )

> compPlot( dat$esdLabMatTL[ dat$monoTL & dat$quasiConcTL ], dat$esaLabMatTL[ dat$monoTL & dat$quasiConcTL ],

+

lim = c( -2, 2 ) )

−2

−1

0

1

2

2 1 0 −2

−1

esaLabMatTL

5 0 −10

−5

esaCapMatTL

0 −1 −2

esaCapLabTL

1

2

10

+

−10

esdCapLabTL

−5

0

5

10

esdCapMatTL

−2

−1

0

1

2

esdLabMatTL

Figure 2.47: Translog production function: Comparison of direct and Allen elasticities of substitution The resulting graphs are shown in figure 2.47. As can also be seen from comparing the histograms in Figures 2.45 and 2.46, the variation of the Allen elasticities of substitution is much larger than the variation of the corresponding direct elasticities of substitution.

2.6.14 Mean-scaled quantities The Translog functional form is often estimated with mean-scaled variables. In the case of a Translog production function, it is estimated with mean-scaled input quantities and sometimes also with mean-scaled output quantities: y ∗ = y/¯ y

(2.143)

x∗i

(2.144)

= xi /¯ xi ∀ i

The following commands create variables with mean-scaled output and input quantities: > dat$qmOut dat$qmCap dat$qmLab dat$qmMat log( colMeans( dat[ , c( "qmOut", "qmCap", "qmLab", "qmMat" ) ] ) )

144

2 Primal Approach: Production Function qmOut

qmCap

qmLab

qmMat

-1.110223e-16 -1.110223e-16

0.000000e+00

0.000000e+00

Please note that mean-scaling does not imply that the mean values of the logarithmic variables are zero: > colMeans( log( dat[ , c( "qmOut", "qmCap", "qmLab", "qmMat" ) ] ) ) qmOut

qmCap

qmLab

qmMat

-0.4860021 -0.3212057 -0.1565112 -0.2128551 The following derivation explores the relationship between a Translog production function with mean-scaled quantities and a Translog production function with the original quantities: ln y ∗ = α0∗ +

X

αi∗ ln x∗i +

i

ln (y/¯ y ) = α0∗ +

X

1 XX ∗ α ln x∗i ln x∗j 2 i j ij 1 XX ∗ α ln (xi /¯ xi ) ln (xj /¯ xj ) 2 i j ij

αi∗ ln (xi /¯ xi ) +

i

ln y − ln y¯ = α0∗ +

X

αi∗ (ln xi − ln x ¯i ) +

i

ln y = α0∗ + ln y¯ +

X

αi∗ ln xi −

(2.147)

αi∗ ln x ¯i

(2.148)

i

XX 1 XX ∗ 1 XX ∗ ∗ αij ln xi ln xj − αij ln xi ln x ¯j + α ln x ¯i ln x ¯j 2 i j 2 i j ij i j 



ln y = α0∗ + ln y¯ −

X i

1 XX ∗ α ln x ¯i ln x ¯j  αi∗ ln x ¯i + 2 i j ij 



+

(2.146)

1 XX ∗ α (ln xi − ln x ¯i ) (ln xj − ln x ¯j ) 2 i j ij

X

i

+

(2.145)

X i

α ∗ i

−

X

∗ αij

ln x ¯j  ln xi +

j

(2.149)

1 XX ∗ α ln xi ln xj 2 i j ij

Thus, the relationship between the coefficients of the Translog function with the original quantities and the coefficients of the Translog function with mean-scaled quantities is: α0 = α0∗ + ln y¯ −

X

αi∗ ln x ¯i +

i

αi = αi∗ −

X

1 XX ∗ α ln x ¯i ln x ¯j 2 i j ij

∗ αij ln x ¯j ∀ i

(2.150) (2.151)

j ∗ αij = αij ∀ i, j.

(2.152)

Accordingly, the reciprocal relationship is: α0∗ = α0 − ln y¯ +

X

αi ln x ¯i +

i

145

1 XX αij ln x ¯i ln x ¯j 2 i j

(2.153)

2 Primal Approach: Production Function αi∗ = αi +

X

αij ln x ¯j ∀ i

(2.154)

j ∗ αij = αij ∀ i, j.

(2.155)

The following command estimates the Translog production function with mean-scaled quantities: > prodTLm summary( prodTLm ) Call: lm(formula = log(qmOut) ~ log(qmCap) + log(qmLab) + log(qmMat) + I(0.5 * log(qmCap)^2) + I(0.5 * log(qmLab)^2) + I(0.5 * log(qmMat)^2) + I(log(qmCap) * log(qmLab)) + I(log(qmCap) * log(qmMat)) + I(log(qmLab) * log(qmMat)), data = dat) Residuals: Min

1Q

Median

3Q

Max

-1.68015 -0.36688

0.05389

0.44125

1.26560

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

-0.09392

0.08815

-1.065

0.28864

log(qmCap)

0.15004

0.11134

1.348

0.18013

log(qmLab)

0.79339

0.17477

4.540 1.27e-05 ***

log(qmMat)

0.50201

0.16608

3.023

I(0.5 * log(qmCap)^2)

-0.02573

0.20834

-0.124

0.90189

I(0.5 * log(qmLab)^2)

-1.16364

0.67943

-1.713

0.08916 .

I(0.5 * log(qmMat)^2)

-0.50368

0.43498

-1.158

0.24902

0.56194

0.29120

1.930

0.05582 .

I(log(qmCap) * log(qmMat)) -0.40996

0.23534

-1.742

0.08387 .

I(log(qmLab) * log(qmMat))

0.42750

1.539

I(log(qmCap) * log(qmLab))

0.65793

0.00302 **

0.12623

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.6412 on 130 degrees of freedom Multiple R-squared:

0.6296,

Adjusted R-squared:

146

0.6039

2 Primal Approach: Production Function F-statistic: 24.55 on 9 and 130 DF,

p-value: < 2.2e-16

As expected, the intercept and the first-order coefficients have adjusted to the new units of measurement, while the second-order coefficients of the Translog function remain unchanged (compare with estimates in section 2.6.2): > all.equal( coef(prodTL)[-c(1:4)], coef(prodTLm)[-c(1:4)], +

check.attributes = FALSE )

[1] TRUE In case of functional forms that are invariant to the units of measurement (e.g. linear, CobbDouglas, quadratic, Translog), mean-scaling does not change the relative indicators of the technology (e.g. output elasticities, elasticities of scale, relative marginal rates of technical substitution, elasticities of substitution). As the logarithms of the mean values of the mean-scaled input quantities are zero, the first-order coefficients are equal to the output elasticities at the sample mean (see equation 2.135), i.e. the output elasticity of capital is 0.15, the output elasticity of labor is 0.793, the output elasticity of materials is 0.502, and the elasticity of scale is 1.445 at the sample mean.

2.6.15 First-order conditions for profit maximization In this section, we will check to what extent the first-order conditions for profit maximization (2.40) are fulfilled, i.e. to what extent the firms use the optimal input quantities. We do this by comparing the marginal value products of the inputs with the corresponding input prices. We can calculate the marginal value products by multiplying the marginal products by the output price: > dat$mvpCapTL dat$mvpLabTL dat$mvpMatTL compPlot( dat$pCap, dat$mvpCapTL ) > compPlot( dat$pLab, dat$mvpLabTL ) > compPlot( dat$pMat, dat$mvpMatTL ) > compPlot( dat$pCap[ dat$monoTL ], dat$mvpCapTL[ dat$monoTL ], log = "xy" ) > compPlot( dat$pLab[ dat$monoTL ], dat$mvpLabTL[ dat$monoTL ], log = "xy" ) > compPlot( dat$pMat[ dat$monoTL ], dat$mvpMatTL[ dat$monoTL ], log = "xy" )

147

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−5

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0

20

40

60

−5

0

5

0

50

w Lab

100

20.00

50 MVP Mat

0.20

10

1.00

MVP Lab

150

w Mat

5.00

5e+01 5e+00 5e−01

0.05

5

5e−02

100

200

w Cap

10 15 20 25 30

20

−40

MVP Cap

50

MVP Mat

15 0

−20

5

10

MVP Lab

20 0

MVP Cap

40

20

60

25

30

2 Primal Approach: Production Function

5e−02

5e−01

5e+00

5e+01

0.05

0.20

w Cap

1.00

5.00 20.00

5

10

w Lab

Figure 2.48: Marginal value products and corresponding input prices

148

20 w Mat

50

100

2 Primal Approach: Production Function The resulting graphs are shown in figure 2.48. They indicate that the marginal value products of most firms are higher than the corresponding input prices. This indicates that most firms could increase their profit by using more of all inputs. Given that the estimated Translog function shows that all firms operate under increasing returns to scale, it is not surprising that most firms would gain from increasing all input quantities. Therefore, the question arises why the firms in the sample did not do this. This questions has already been addressed in section 2.3.10.

2.6.16 First-order conditions for cost minimization As the marginal rates of technical substitution differ between observations for the three other functional forms, we use scatter plots for visualizing the comparison of the input price ratios with the negative inverse marginal rates of technical substitution: As the marginal rates of technical substitution are meaningless if the monotonicity condition is not fulfilled, we limit the comparisons to the observations, where all monotonicity conditions are fulfilled: > compPlot( ( dat$pCap / dat$pLab )[ dat$monoTL ], +

- dat$mrtsLabCapTL[ dat$monoTL ] )

> compPlot( ( dat$pCap / dat$pMat )[ dat$monoTL ], +

- dat$mrtsMatCapTL[ dat$monoTL ] )

> compPlot( ( dat$pLab / dat$pMat )[ dat$monoTL ], +

- dat$mrtsMatLabTL[ dat$monoTL ] )

> compPlot( ( dat$pCap / dat$pLab )[ dat$monoTL ], +

- dat$mrtsLabCapTL[ dat$monoTL ], log = "xy" )

> compPlot( ( dat$pCap / dat$pMat )[ dat$monoTL ], +

- dat$mrtsMatCapTL[ dat$monoTL ], log = "xy" )

> compPlot( ( dat$pLab / dat$pMat )[ dat$monoTL ], +

- dat$mrtsMatLabTL[ dat$monoTL ], log = "xy" )

The resulting graphs are shown in figure 2.49. Furthermore, we use histograms to visualize the (absolute and relative) differences between the input price ratios and the corresponding negative inverse marginal rates of technical substitution: > hist( ( - dat$mrtsLabCapTL - dat$pCap / dat$pLab )[ dat$monoTL ] ) > hist( ( - dat$mrtsMatCapTL - dat$pCap / dat$pMat )[ dat$monoTL ] ) > hist( ( - dat$mrtsMatLabTL - dat$pLab / dat$pMat )[ dat$monoTL ] ) > hist( log( - dat$mrtsLabCapTL / ( dat$pCap / dat$pLab ) )[ dat$monoTL ] ) > hist( log( - dat$mrtsMatCapTL / ( dat$pCap / dat$pMat ) )[ dat$monoTL ] ) > hist( log( - dat$mrtsMatLabTL / ( dat$pLab / dat$pMat ) )[ dat$monoTL ] ) The resulting graphs are shown in figure 2.50. The graphs in the middle column of figures 2.49 and 2.50 show that the ratio between the capital price and the materials price is larger than the

149

60

3 0.0 0.1 0.2 0.3 0.4 0.5 0.6 w Cap / w Mat

w Cap / w Lab

1e+02

3

0.001

0.001 1e+00

2

1.000 − MRTS Mat Lab

0.100 0.020 0.005

− MRTS Mat Cap

1e+01 1e+00 1e−01 1e−02

1

w Lab / w Mat

0.500

1e+02

w Cap / w Lab

0

0.100

40

1e−02

− MRTS Lab Cap

20

0.010

0

0

0

0.0

0.1

1

2

− MRTS Mat Lab

0.4 0.3 0.2

− MRTS Mat Cap

40 20

− MRTS Lab Cap

60

0.5

0.6

4

2 Primal Approach: Production Function

0.001

0.005

0.050

0.500

0.001

w Cap / w Mat

Figure 2.49: First-order conditions for costs minimization

150

0.010

0.100

w Lab / w Mat

1.000

4

60

80

10 −0.6

0.2

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0.6

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− MrtsMatCap − wCap / wMat

1

2

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4

− MrtsMatLab − wLab / wMat

20 15 10

Frequency

10

20

Frequency

15 10

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log(−MrtsLabCap / (wCap / wLab))

0

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5 0

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− MrtsLabCap − wCap / wLab

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0

5 0 20

40 20

10 10 0

0

30

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20 15

Frequency

30 20

Frequency

25

50

40

30

60

2 Primal Approach: Production Function

−4

−2

0

2

log(−MrtsMatCap / (wCap / wMat))

−6

−2

0

2

log(−MrtsMatLab / (wLab / wMat))

Figure 2.50: First-order conditions for costs minimization

151

−4

2 Primal Approach: Production Function absolute value of the marginal rate of technical substitution between materials and capital for a majority of the firms in the sample: wcap M Pcap > −M RT Smat,cap = wmat M Pmat

(2.156)

Hence, these firms can get closer to the minimum of their production costs by substituting materials for capital, because this will decrease the marginal product of materials and increase the marginal product of capital so that the absolute value of the MRTS between materials and capital increases and gets closer to the corresponding input price ratio. The graphs on the left indicate that approximately half of the firms should substitute labor for capital, while the other half should substitute capital for labor. The graphs on the right indicate that a majority of the firms should substitute materials for labor. Hence, the majority of the firms could reduce production costs particularly by using more materials and using less labor or less capital but there might be (legal) regulations that restrict the use of materials (e.g. fertilizers, pesticides).

2.7 Evaluation of different functional forms In this section, we will discuss the appropriateness of the four different functional forms for analyzing the production technology in our data set. If one functional form is nested in another functional form, we can use standard statistical tests to compare these functional forms. We have done this already in section 2.5 (linear production function vs. quadratic production function) and in section 2.6 (Cobb-Douglas production function vs. Translog production function). The tests clearly reject the linear production function in favor of the quadratic production function but it is less clear whether the Cobb-Douglas production function is rejected in favor of the Translog production function. It is much less straight-forward to compare non-nested models such as the quadratic and the Translog production functions.

2.7.1 Goodness of fit As the linear and quadratic models use different dependent variables than the Cobb-Douglas and Translog models (y vs. ln y), we cannot simply compare the R2 -values. However, we can calculate the hypothetical R2 -values regarding y for the Cobb-Douglas and Translog production functions and compare them with the R2 values of the linear and quadratic production functions. We can also calculate the hypothetical R2 -value regarding ln y for the linear and quadratic production functions and compare them with the R2 values of the Cobb-Douglas and Translog production functions. We can calculate the (hypothetical) R2 values with function rSquared (package miscTools). The first argument of this function must be a vector of the observed dependent variable and the second argument must be a vector of the residuals. We start by extracting the R2 values from the linear and quadratic models and calculate the hypothetical R2 -values regarding y for

152

2 Primal Approach: Production Function the Cobb-Douglas and Translog production functions: > summary(prodLin)$r.squared [1] 0.786819 > summary(prodQuad)$r.squared [1] 0.8448983 > rSquared( dat$qOut, dat$qOut - dat$qOutCD ) [,1] [1,] 0.8067198 > rSquared( dat$qOut, dat$qOut - dat$qOutTL ) [,1] [1,] 0.7696638 In this case, the R2 value regarding y is considerably higher for the quadratic function than for the other functional forms, particularly the Translog functional form, which has the lowest R2 value regarding y. Similarly, we can extract the R2 values from the Cobb-Douglas and Translog models and calculate the hypothetical R2 -values regarding ln y for the linear and quadratic production functions. As the logarithm of non-positive numbers is not defined, we need to exclude observations with non-positive predicted output quantities: > summary(prodCD)$r.squared [1] 0.5943154 > summary(prodTL)$r.squared [1] 0.6295696 > rSquared( log( dat$qOut[ dat$qOutLin > 0 ] ), +

log( dat$qOut[ dat$qOutLin > 0 ] ) - log( dat$qOutLin[ dat$qOutLin > 0 ] ) ) [,1]

[1,] 0.3796727 > rSquared( log( dat$qOut ), log( dat$qOut ) - log( dat$qOutQuad ) ) [,1] [1,] 0.5481309

153

2 Primal Approach: Production Function In contrast to the R2 values regarding y, the R2 value regarding ln y is considerably higher for the Translog function than for the other functional forms. Hence, in our case, the R2 values do not help much to select the most suitable functional form. As the quadratic and the Translog functional forms have the same number of coefficients, we can base our comparison of these functional forms on the unadjusted R2 values. In order to compare models that have different numbers of coefficients, one should rather use adjusted R2 values. Furthermore, we can visually compare the fit of the two models by looking at figures 2.7, 2.15, 2.24 and 2.37. The Cobb-Douglas and quadratic production functions clearly over-predict the output of small firms so that small firms have rather large relative error terms. On the other hand, the Translog production function has rather large absolute error terms for large firms. In total, it seems that the fit of the linear and Translog production functions is slightly better than the fit of the Cobb-Douglas and quadratic production functions.

2.7.2 Test for functional form misspecification We conduct Ramsey’s (1969) Regression Equation Specification Error Test (RESET) on all four functional forms: > resettest( prodLin ) RESET test data:

prodLin

RESET = 17.639, df1 = 2, df2 = 134, p-value = 1.584e-07 > resettest( prodCD ) RESET test data:

prodCD

RESET = 2.9224, df1 = 2, df2 = 134, p-value = 0.05724 > resettest( prodQuad ) RESET test data:

prodQuad

RESET = 7.3663, df1 = 2, df2 = 128, p-value = 0.0009374 > resettest( prodTL ) RESET test data:

prodTL

RESET = 1.2811, df1 = 2, df2 = 128, p-value = 0.2813

154

2 Primal Approach: Production Function While the linear and quadratic functional forms are clearly rejected, the Cobb-Douglas functional form is only rejected at the 10%, and the Translog is not rejected at all.

2.7.3 Theoretical consistency Furthermore, we can compare the theoretical consistency of the models. The total number of monotonicity violations of the four functional forms can be obtained by > with( dat, sum( eCap < 0 ) + sum( eLab < 0 ) + sum( eMat < 0 ) ) [1] 0 > with( dat, sum( mpCapCD < 0 ) + sum( mpLabCD < 0 ) + sum( mpMatCD < 0 ) ) [1] 0 > with( dat, sum( eCapQuad < 0 ) + sum( eLabQuad < 0 ) + sum( eMatQuad < 0 ) ) [1] 41 > with( dat, sum( eCapTL < 0 ) + sum( eLabTL < 0 ) + sum( eMatTL < 0 ) ) [1] 54 Alternatively, we could look at the number of observations, at which the monotonicity condition is violated: > table( dat$monoQuad ) FALSE

TRUE

39

101

> table( dat$monoTL ) FALSE

TRUE

48

92

While the monotonicity conditions are globally fulfilled for the linear and Cobb-Douglas functional forms, the more flexible functional forms have several monotonicity violations. When comparing only the two more flexible functional forms, we see that the monotonicity condition is more often violated in the Translog function than in the quadratic function (according to both of the measures of monotonicity violation). While the Cobb-Douglas and Translog production functions always return positive output quantities (as long as all input quantities are strictly positive), this is not necessarily the case for linear and quadratic production functions. However, we have checked this in sections 2.3.4

155

2 Primal Approach: Production Function and 2.5.6 and found only one negative predicted output quantity for the linear production function, while all output quantities predicted by the quadratic production function are positive. Hence, the non-negativity condition is almost fulfilled for the linear production function, while it is completely fulfilled for the remaining three functional forms. Quasiconcavity is fulfilled at all observations for the linear and Cobb-Douglas production functions and at 63 out of 140 observations for the Translog production function, while it is fulfilled at no observation for the quadratic production function. However, quasiconcavity is mainly assumed to simplify the (further) economic analysis (e.g. to obtain continuous input demand and output supply functions) and there can be found good reasons for why the true production technology is not quasiconcave (e.g. indivisibility of inputs).

2.7.4 Plausible estimates While the elasticities of scale of some observations were implausibly large when estimated with the linear production function, no elasticities of scale estimated by the Cobb-Douglas, quadratic and Translog production functions are in the implausible range: > table( dat$eScale >= 0.5 & dat$eScale table( rep( sum( coef( prodCD )[-1] ) >= 0.5 & sum( coef( prodCD )[-1] ) table( dat$eScaleQuad >= 0.5 & dat$eScaleQuad table( dat$eScaleTL >= 0.5 & dat$eScaleTL with( dat, sum( eCap > 1 ) + sum( eLab > 1 ) + sum( eMat > 1 ) ) [1] 124

156

2 Primal Approach: Production Function > ( coef( prodCD )[2] > 1 ) * nrow( dat ) + +

( coef( prodCD )[3] > 1 ) * nrow( dat ) +

+

( coef( prodCD )[4] > 1 ) * nrow( dat )

log(qCap) 0 > with( dat, sum( eCapQuad > 1 ) + sum( eLabQuad > 1 ) + sum( eMatQuad > 1 ) ) [1] 28 > with( dat, sum( eCapTL > 1 ) + sum( eLabTL > 1 ) + sum( eMatTL > 1 ) ) [1] 56 The linear production function indicates the most implausibly large output elasticities, followed by the Translog, quadratic, and Cobb-Douglas production functions, whereas the Cobb-Douglas production function has no implausibly large output elasticities. When comparing just the two more flexible functional forms, the Translog production function results in twice as many implausible output elasticities than the quadratic production function.

2.7.5 Summary The various criteria for assessing and comparing the appropriateness of the four functional forms for analyzing the production technology in our data set are summarized in table 2.2. While the linear production function seems to be the least appropriate for our specific empirical analysis, the Cobb-Douglas functions performs rather well according to most indicators. When comparing the two more flexible functional forms, we can see that the quadratic production function results in less monotonicity violations and less implausible output elasticities than the Translog production function, while the Translog production function seems to give a better fit to the data (visually and according to Ramsey’s RESET).

2.8 Non-parametric production function In order to avoid the specification of a functional form of the production function, the production technology can be analyzed by nonparametric regression. We will use a local-linear kernel regressor with an Epanechnikov kernel for the (continuous) regressors (see, e.g. Li and Racine, 2007; Racine, 2008). “One can think of this estimator as a set of weighted linear regressions, where a weighted linear regression is performed at each observation and the weights of the other observations decrease with the distance from the respective observation. The weights are determined by a kernel function and a set of bandwidths, where a bandwidth for each explanatory variable must be specified. The smaller the bandwidth, the faster the weight decreases with the distance

157

2 Primal Approach: Production Function

Table 2.2: Criteria for assessing functional forms linear Cobb-Douglas quadratic 2 R of y 0.79 0.81 0.84 R2 of ln y 0.38 0.59 0.55 visual fit + − − RESET (P-value) 0 0.05724 0.00094 total monotonicity violations 0 0 41 observations with monotonicity violated 0 0 39 negative output quantities 1 0 0 observations with quasiconcavity violated 0 0 140 implausible elasticities of scale 67 0 0 implausible output elasticities 124 0 28

Translog 0.77 0.63 (+) 0.28127 54 48 0 77 0 56

from the respective observation. In our study, we make the frequently used assumption that the bandwidths can differ between regressors but are constant over the domain of each regressor. While the bandwidths were initially determined by using a rule of thumb, nowadays increased computing power allows us to select the optimal bandwidths for a given model and data set according to the expected Kullback-Leibler cross-validation criterion (Hurvich, Simonoff, and Tsai, 1998). Hence, in nonparametric kernel regression, the overall shape of the relationship between the inputs and the output is determined by the data and the (marginal) effects of the explanatory variables can differ between observations without being restricted by an arbitrarily chosen functional form.” (Czekaj and Henningsen, 2012). Given that the distributions of the output quantity and the input quantities are strongly right-skewed in our data set (many firms with small quantities, only a few firms with large quantities), we use the logarithms of the output and input quantities in order to achieve more uniform distributions, which are preferable in case of fixed bandwidths. Furthermore, this allows us to interpret the gradients of the dependent variable (logarithmic output quantity) with respect to the explanatory variables (logarithmic input quantities) as output elasticities. The following commands load the R package np (Hayfield and Racine, 2008), select the optimal bandwidths and estimate the model, and show summary results: > library( "np" ) > prodNP summary( prodNP ) Regression Data: 140 training points, in 3 variable(s) log(qCap) log(qLab) log(qMat) Bandwidth(s):

1.039647

332644 0.8418465

Kernel Regression Estimator: Local-Linear

158

2 Primal Approach: Production Function Bandwidth Type: Fixed Residual standard error: 0.6227669 R-squared: 0.6237078 Continuous Kernel Type: Second-Order Epanechnikov No. Continuous Explanatory Vars.: 3 While the bandwidths of the logarithmic quantities of capital and materials are around one, the bandwidth of the logarithmic labor quantity is rather large. These bandwidths indicate that the logarithmic output quantity non-linearly changes with the logarithmic quantities of capital and materials but it changes approximately linearly with the logarithmic labor quantity. The estimated relationship between each explanatory variable and the dependent variable (holding all other explanatory variables constant at their median values) can be visualized using the plot method. We can use argument plot.errors.method to add confidence intervals: > plot( prodNP, plot.errors.method = "bootstrap" ) The resulting graphs are shown in figure 2.51. The estimated gradients of the dependent variable with respect to each explanatory variable (holding all other explanatory variables constant at their median values) can be visualized using the plot method with argument gradient set to TRUE: > plot( prodNP, gradients = TRUE, plot.errors.method = "bootstrap" ) The resulting graphs are shown in figure 2.52. Function npsigtest can be used to obtain the statistical significance of the explanatory variables: > npsigtest( prodNP ) Kernel Regression Significance Test Type I Test with IID Bootstrap (399 replications, Pivot = TRUE, joint = FALSE) Explanatory variables tested for significance: log(qCap) (1), log(qLab) (2), log(qMat) (3) log(qCap) log(qLab) log(qMat) Bandwidth(s):

1.039647

332644 0.8418465

Individual Significance Tests P Value: log(qCap) 0.12281 log(qLab) < 2e-16 *** log(qMat) < 2e-16 ***

159

15 14 13

log(qOut)

16

2 Primal Approach: Production Function

9

10

11

12

13

15 14 13

log(qOut)

16

log(qCap)

11.5

12.0

12.5

13.0

13.5

14.0

15 14 13

log(qOut)

16

log(qLab)

9.0

9.5

10.0

10.5

11.0

11.5

log(qMat)

Figure 2.51: Production technology estimated by non-parametric kernel regression 160

1.0 0.5 0.0 −0.5

Gradient Component 1 of log(qOut)

2 Primal Approach: Production Function

9

10

11

12

13

1.0 0.5 0.0 −0.5

Gradient Component 2 of log(qOut)

log(qCap)

11.5

12.0

12.5

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14.0

1.0 0.5 0.0 −0.5

Gradient Component 3 of log(qOut)

log(qLab)

9.0

9.5

10.0

10.5

11.0

11.5

log(qMat)

Figure 2.52: Gradients (output elasticities) estimated by non-parametric kernel regression 161

2 Primal Approach: Production Function --Signif. codes:

0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1

The results confirm the results from the parametric regressions that labor and materials have a significant effect on the output while capital does not have a significant effect (at 10% significance level). The following commands plot histograms of the three output elasticities and the elasticity of scale: > hist( gradients( prodNP )[ ,1] ) > hist( gradients( prodNP )[ ,2] ) > hist( gradients( prodNP )[ ,3] )

40 0

20

Frequency

40 20 0

Frequency

> hist( rowSums( gradients( prodNP ) ) )

−0.1

0.0

0.1

0.2

0.3

0.0

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1.0

labor

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20 0

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40

capital

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1.3

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1.5

materials

1.6

1.7

1.8

1.9

scale

Figure 2.53: Output elasticities and elasticities of scale estimated by non-parametric kernel regression The resulting graphs are shown in figure 2.53. The monotonicity condition is fulfilled at almost all observations, only 1 output elasticity of capital and 0 output elasticity of labor is negative. All firms operate under increasing returns to scale with most farms having an elasticity of scale around 1.4. Finally, we visualize the relationship between firm size and the elasticity of scale based on our non-parametric estimation results: > plot( dat$qOut, rowSums( gradients( prodNP ) ), log = "x" ) > plot( dat$X, rowSums( gradients( prodNP ) ), log = "x" )

162

1.9 1.7 1.3

1.5

elaScaleNP

1.7 1.5 1.3

elaScaleNP

1.9

2 Primal Approach: Production Function

1e+05

5e+05

5e+06

0.5

qOut

1.0

2.0

5.0

X

Figure 2.54: Relationship between firm size and elasticities of scale estimated by non-parametric kernel regression The resulting graph is shown in figure 2.54. The smallest firms generally would gain most from increasing their size. However, also the largest firms would still considerably gain from increasing their size—perhaps even more than medium-sized firms but there is probably insufficient evidence to be sure about this.

163

3 Dual Approach: Cost Functions 3.1 Theory 3.1.1 Cost function Total cost is defined as: c=

X

wi x i

(3.1)

i

The cost function: c(w, y) = min x

X

wi xi , s.t. f (x) ≥ y

(3.2)

i

returns the minimal (total) cost that is required to produce at least the output quantity y given input prices w. It is important to distinguish the cost definition (3.1) from the cost function (3.2).

3.1.2 Properties of the cost function It is usually assumed that a cost function c(w, y) defined in (3.2) fulfills the following properties (see, e.g., Chambers, 1988, p. 51-56): 1. c(w, y) is non-negative, i.e., c(w, y) ≥ 0 ∀ w ≥ 0 and y ≥ 0 2. c(w, y) is non-decreasing in w, i.e. ∂c(w, y)/∂wi ≥ 0 ∀ i 3. c(w, y) is linearly homogeneous in w, i.e. c(k w, y) = k c(w, y) ∀ k > 0 4. c(w, y) is concave in w 5. c(w, y) is non-decreasing in y, i.e. ∂c(w, y)/∂y ≥ 0 6. no fixed costs, i.e., c(w, 0) = 0

3.1.3 Cost flexibility and elasticity of size The ratio between the relative change in total costs and the relative change in the output quantity is called “cost flexibility:” ∂c(w, y) y ∂y c(w, y)

164

(3.3)

3 Dual Approach: Cost Functions The inverse of the cost flexibility is called “elasticity of size:” ∗ (w, y) =

c(w, y) ∂y ∂c(w, y) y

(3.4)

At the cost-minimizing points, the elasticity of size is equal to the elasticity of scale (Chambers, 1988, p. 71–72). For homothetic production technologies such as the Cobb-Douglas production technology, the elasticity of size is always equal to the elasticity of scale (Chambers, 1988, p. 72– 74).1

3.1.4 Homotheticity of cost functions A cost function is homothetic if it can be decomposed as: c(w, y) = c0 (c1 (w), c2 (y)),

(3.5)

where c1 (w) is a function that aggregates the input prices (w) to a single value, c2 (y) is a function that transforms the output quantity (y) (or—in the case of multiple outputs—that aggregates the output quantities to a single value), and c0 (c1 , c2 ) is a function that returns the costs based on the aggregated input price and the transformed output quantity. If a cost function is homothetic, there are no interactions between individual input prices and the output quantity, i.e., when the output quantity changes, the cost function is just scaled up or scaled down (proportionally or non-proportionally with the output quantity), while its ‘shape’ remains unchanged. This means, for example, that the optimal cost shares of the inputs are independent of the scale of production (i.e., they do not depend on the output quantities).

3.1.5 Short-run cost functions As producers often cannot instantly adjust the quantity of the some inputs (e.g. buildings, land, 0

0

apple trees), we separate the vector of input quantities (x) into two subvectors (x = (x1 , x2 )0 ), where vector x1 includes the quantities of the variable inputs, i.e. the inputs that can be freely adjusted in the short run, and vector x2 includes the quantities of the quasi-fixed inputs, i.e. the inputs that cannot be adjusted in the short run. In this case, the production function becomes y = f (x1 , x2 ). In the long run, when all input quantities can be freely adjusted, the cost minimization problem remains: 0

0

min w1 x1 + w2 x2 , s.t. f (x1 , x2 ) ≥ y

x1 ,x2

1

(3.6)

Further details about the relationship between the elasticity of size and the elasticity of scale are available, e.g., in McClelland, Wetzstein, and Musserwetz (1986).

165

3 Dual Approach: Cost Functions and the cost function remains 0

0

c(w, y) = c(w1 , w2 , y) = min w1 x1 + w2 x2 , s.t. f (x1 , x2 ) ≥ y, x1 ,x2

(3.7)

where w1 denotes the vector of the prices of the variable inputs and w2 denotes the vector of the prices of the quasi-fixed inputs. However, in the short run, when only the variable input quantities (x1 ) can be freely adjusted, while the quasi-fixed input quantities (x2 ) cannot be adjusted, the costs of the quasi-fixed inputs 0

(w2 x2 ) are “sunk costs” and thus are irrelevant for short-run cost minimization. Hence, one only needs to take into account—and minimize—the variable costs: 0

min w1 x1 , s.t. f (x1 , x2 ) ≥ y.

(3.8)

x1

Based on this short-run (variable) cost minimization, we can derive the short-run (variable) cost function: 0

cv (w1 , y, x2 ) = min w1 x1 , s.t. f (x1 , x2 ) ≥ y,

(3.9)

x1

where cv denotes the variable costs defined in equation (1.3). Total short-run costs consist of variable costs and sunk costs (see, e.g., Chambers, 1988, p. 103): 0

cs (w, y, x2 ) = cs (w1 , w2 , y, x2 ) = cv (w1 , y, x2 ) + w2 x2 .

(3.10)

The long-run cost function can be expressed in terms of the short-run total cost function or the short-run (variable) cost function (see, e.g., Chambers, 1988, p. 103): h

0

i

c(w, y) = min cs (w1 , w2 , y, x2 ) = min cv (w1 , y, x2 ) + w2 x2 . x2

x2

(3.11)

If producers cannot instantly adjust the quantity of the some inputs, estimating a short-run cost function with some quasi-fixed input quantities might be more appropriate than estimating a (long-run) cost function which assumes that all input quantities quantities can be adjusted instantly.

3.1.6 Cost functions with multiple outputs The cost function with multiple outputs is very similar to the cost function with a single output (3.2): c(w, y) = min x

X

wi xi , s.t. (x, y) ∈ T,

(3.12)

i

where w = (w1 , . . . , wN )0 and x = (x1 , . . . , xN )0 are vectors of input prices and inputs quantities, respectively, as defined before, T is again the technology set, and y = (y1 , . . . , yM )0 is a vector of M output quantities. The multiple-output cost function c(w, y) returns the minimal (total) cost that is required to produce the output quantities y = (y1 , . . . , yM )0 given input prices w =

166

3 Dual Approach: Cost Functions (w1 , . . . , wN )0 . The properties of the multiple-output cost function are basically the same as the properties of the single-output cost function that are described in section 3.1.2. However, the monotonicity property regarding the output quantity applies to all output quantities, i.e.: ˆ c(w, y) is non-decreasing in all y, i.e. ∂c(w, y)/∂ym ≥ 0 ∀ m = 1, . . . , M

and it has one additional property: ˆ c(w, y) is quasi-convex in output quantities ⇒ concave iso-cost curves for different output-

output combinations. In case of multiple outputs, the cost flexibility is: M X ∂c(w, y) m=1

∂ym

ym . c(w, y)

(3.13)

It indicates by how many percent the (minimum) total cost changes if all output quantities are increased by one percent. As in the single-input case, the elasticity of size is the inverse of the cost flexibility: ∗ (w, y) =

M X ∂c(w, y) m=1

∂ym

ym c(w, y)

!−1

.

(3.14)

It indicates by how many percent all outputs can be (proportionally) increased if the (minimum) total cost is increased by one percent.

3.1.7 Suitability of the cost function for econometric applications Given the microeconomic theory of the cost function and the assumptions of standard econometric methods such as ordinary least squares (OLS), it is appropriate to estimate a cost function with these econometric methods if several conditions are fulfilled. The most important and relevant conditions are: 1. The inputs and outputs are rather similar across all firms in the data set (within each input or output category). 2. The production conditions are rather similar for all firms in the data set (unless the empirical specification appropriately accounts for these differences). 3. All firms in the data set minimize their costs to produce the given output quantities given the input prices. 4. All input quantities must be “variable”, i.e., must be adjustable by the producer from one observational period to the next observational period (e.g., from year to year in case of annual data).

167

3 Dual Approach: Cost Functions 5. No firm in the data set has all output quantities equal to zero but has strictly positive costs (i.e., the “no fixed costs” assumption must be fulfilled at all observations). 6. All output quantities and input prices (i.e., all explanatory variables) are uncorrelated with the error term, which is fulfilled, e.g., if the output quantities are exogenously given (e.g., by the demand for these outputs if the output prices are exogeneously given or by a non-tradable quota for the outputs) and the firms have no market power and, thus, are price-takers on the input markets.

3.2 Cobb-Douglas cost function 3.2.1 Specification We start with estimating a Cobb-Douglas cost function. It has the following specification: ! Y α c=A wi i

y αy

(3.15)

αi ln wi + αy ln y

(3.16)

i

This function can be linearized to ln c = α0 +

X i

with α0 = ln A.

3.2.2 Estimation The linearized Cobb-Douglas cost function can be estimated by OLS: > costCD summary( costCD ) Call: lm(formula = log(cost) ~ log(pCap) + log(pLab) + log(pMat) + log(qOut), data = dat) Residuals: Min

1Q

Median

3Q

Max

-0.77663 -0.23243 -0.00031

0.24439

0.74339

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

6.75383

0.40673

16.605

< 2e-16 ***

168

3 Dual Approach: Cost Functions log(pCap)

0.07437

0.04878

1.525

0.12969

log(pLab)

0.46486

0.14694

3.164

0.00193 **

log(pMat)

0.48642

0.08112

5.996 1.74e-08 ***

log(qOut)

0.37341

0.03072

12.154

< 2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3395 on 135 degrees of freedom Multiple R-squared:

0.6884,

Adjusted R-squared:

F-statistic: 74.56 on 4 and 135 DF,

0.6792

p-value: < 2.2e-16

3.2.3 Properties As the coefficients of the (logarithmic) input prices are all non-negative, this cost function is monotonically non-decreasing in input prices. Furthermore, the coefficient of the (logarithmic) output quantity is non-negative so that this cost function is monotonically non-decreasing in output quantities. The Cobb-Douglas cost function always implies no fixed costs, as the costs are always zero if the output quantity is zero. Given that A = exp(α0 ) is always positive, all Cobb-Douglas cost functions that are based on its (estimated) linearized version fulfill the non-negativity condition. Finally, we check if the Cobb-Douglas cost function is positive linearly homogeneous in input prices. This condition is fulfilled if t c(w, y) = c(t w, y) ln(t c) = α0 +

X

ln t + ln c = α0 +

X

(3.17) αi ln(t wi ) + αy ln y

(3.18)

X

(3.19)

i

αi ln t +

i

αi ln wi + αy ln y

i

ln c + ln t = α0 + ln t

X

αi +

i

ln c + ln t = ln c + ln t

X

αi ln wi + αy ln y

(3.20)

i

X

αi

(3.21)

i

ln t = ln t

X

αi

(3.22)

i

1=

X

αi

(3.23)

i

Hence, the homogeneity condition is only fulfilled if the coefficients of the (logarithmic) input prices sum up to one. As they sum up to 1.03 the homogeneity condition is not fulfilled in our estimated model.

169

3 Dual Approach: Cost Functions

3.2.4 Estimation with linear homogeneity in input prices imposed In order to estimate a Cobb-Douglas cost function with linear homogeneity imposed, we re-arrange the homogeneity condition to get αN = 1 −

N −1 X

αi

(3.24)

i=1

and replace αN in the cost function (3.16) by the right-hand side of the above equation: ln c = α0 + ln c = α0 + ln c − ln wN = α0 + ln

c = α0 + wN

N −1 X i=1 N −1 X i=1 N −1 X i=1 N −1 X

αi ln wi + 1 −

N −1 X

!

αi ln wN + αy ln y

(3.25)

i=1

αi (ln wi − ln wN ) + ln wN + αy ln y

(3.26)

αi (ln wi − ln wN ) + αy ln y

(3.27)

αi ln

i=1

wi + αy ln y wN

(3.28)

This Cobb-Douglas cost function with linear homogeneity in input prices imposed can be estimated by following command: > costCDHom summary( costCDHom ) Call: lm(formula = log(cost/pMat) ~ log(pCap/pMat) + log(pLab/pMat) + log(qOut), data = dat) Residuals: Min

1Q

Median

3Q

Max

-0.77096 -0.23022 -0.00154

0.24470

0.74688

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

6.75288

0.40522

16.665

log(pCap/pMat)

0.07241

0.04683

1.546

log(pLab/pMat)

0.44642

0.07949

5.616 1.06e-07 ***

log(qOut)

0.37415

0.03021

12.384

< 2e-16 *** 0.124 < 2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

170

3 Dual Approach: Cost Functions Residual standard error: 0.3383 on 136 degrees of freedom Multiple R-squared:

0.5456,

Adjusted R-squared:

F-statistic: 54.42 on 3 and 136 DF,

0.5355

p-value: < 2.2e-16

The coefficient of the N th (logarithmic) input price can be obtained by the homogeneity condition (3.24). Hence, the estimate of αMat is 0.4812 in our model. As there is no theory that says which input price should be taken for the normalization/deflation, it is desirable that the estimation results do not depend on the price that is used for the normalization/deflation. This desirable property is fulfilled for the Cobb-Douglas cost function and we can verify this by re-estimating the cost function, while using a different input price for the normalization/deflation, e.g. capital: > costCDHomCap summary( costCDHomCap ) Call: lm(formula = log(cost/pCap) ~ log(pLab/pCap) + log(pMat/pCap) + log(qOut), data = dat) Residuals: Min

1Q

Median

3Q

Max

-0.77096 -0.23022 -0.00154

0.24470

0.74688

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

6.75288

0.40522

16.665

log(pLab/pCap)

0.44642

0.07949

5.616 1.06e-07 ***

log(pMat/pCap)

0.48117

0.07285

6.604 8.26e-10 ***

log(qOut)

0.37415

0.03021

12.384

< 2e-16 ***

< 2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3383 on 136 degrees of freedom Multiple R-squared:

0.8168,

F-statistic: 202.2 on 3 and 136 DF,

Adjusted R-squared:

0.8128

p-value: < 2.2e-16

The results are identical to the results from the Cobb-Douglas cost function with the price of materials used for the normalization/deflation. The coefficient of the (logarithmic) capital price can be obtained by the homogeneity condition (3.24). Hence, the estimate of αCap is 0.0724 in our model with the capital price as num´eraire, which is identical to the corresponding estimate from the model with the price of materials as num´eraire. Both models have identical residuals:

171

3 Dual Approach: Cost Functions > all.equal( residuals( costCDHom ), residuals( costCDHomCap ) ) [1] TRUE However, as the two models have different dependent variables (c/pMat and c/pCap ), the R2 -values differ between the two models. We can test the restriction for imposing linear homogeneity in input prices, e.g. by a Wald test or a likelihood ratio test. As the models without and with homogeneity imposed (costCD and costCDHom) have different dependent variables (c and c/pMat ), we cannot use the function waldtest for conducting the Wald test but we have to use the function linearHypothesis (package car) and specify the homogeneity restriction manually: > library( "car" ) > linearHypothesis( costCD, "log(pCap) + log(pLab) + log(pMat) = 1"

)

Linear hypothesis test Hypothesis: log(pCap)

+ log(pLab)

+ log(pMat) = 1

Model 1: restricted model Model 2: log(cost) ~ log(pCap) + log(pLab) + log(pMat) + log(qOut) Res.Df

RSS Df Sum of Sq

1

136 15.563

2

135 15.560

F Pr(>F)

1 0.0025751 0.0223 0.8814

> lrtest( costCDHom, costCD ) Likelihood ratio test Model 1: log(cost/pMat) ~ log(pCap/pMat) + log(pLab/pMat) + log(qOut) Model 2: log(cost) ~ log(pCap) + log(pLab) + log(pMat) + log(qOut) #Df

LogLik Df

1

5 -44.878

2

6 -44.867

Chisq Pr(>Chisq)

1 0.0232

0.879

These two tests clearly show that the data do not contradict linear homogeneity in input prices. Alternatively, one can test the homogeneity restriction by rearranging the hypothesis 1 to δ = 0 with δ =

PN

i=1 αi

PN

i=1 αi

=

− 1. The definition of the auxiliary variable δ can be re-arranged to: αN = 1 −

N −1 X i=1

172

αi + δ

(3.29)

3 Dual Approach: Cost Functions Replacing αN in equation (3.16) by the right-hand side of equation (3.29), we get (similarly to the derivation in equations (3.25)–(3.28)): N −1 X wi c = α0 + αi ln + δ ln wN + αy ln y ln wN w N i=1

(3.30)

This equation can be estimated by OLS and a standard t-test can be used to test the restriction δ = 0: > summary( lm( log( cost / pMat ) ~ log( pCap / pMat ) + log( pLab / pMat ) + +

log( pMat ) + log( qOut ), data = dat ) )

Call: lm(formula = log(cost/pMat) ~ log(pCap/pMat) + log(pLab/pMat) + log(pMat) + log(qOut), data = dat) Residuals: Min

1Q

Median

3Q

Max

-0.77663 -0.23243 -0.00031

0.24439

0.74339

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

6.75383

0.40673

16.605

< 2e-16 ***

log(pCap/pMat)

0.07437

0.04878

1.525

0.12969

log(pLab/pMat)

0.46486

0.14694

3.164

0.00193 **

log(pMat)

0.02565

0.17159

0.149

0.88140

log(qOut)

0.37341

0.03072

12.154

< 2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3395 on 135 degrees of freedom Multiple R-squared:

0.5456,

F-statistic: 40.53 on 4 and 135 DF,

Adjusted R-squared:

0.5322

p-value: < 2.2e-16

This test is identical to the Wald test conducted above and hence, gives exactly the same P-value.

3.2.5 Checking concavity in input prices The last property that we have to check is the concavity in input prices. A continuous and twice continuously differentiable function is concave, if its Hessian matrix is negative semidefinite. A necessary and sufficient condition for negative semidefiniteness of a symmetric matrix is that all its ith-order principal minors (not only its leading principal minors) are non-positive for i

173

3 Dual Approach: Cost Functions being odd and non-negative for i being even for all i ∈ {1, . . . , N } (see section 1.5.2). The first derivatives of the Cobb-Douglas cost function with respect to the input prices are: c ∂c ∂ ln c c = = αi ∂wi ∂ ln wi wi wi

(3.31)

Now, we can calculate the second derivatives as derivatives of the first derivatives (3.31): 



∂c ∂ αi wci ∂ ∂w ∂2c i = = ∂wi ∂wj ∂wj ∂wj c αi ∂c − δij αi 2 = wi ∂wj wi c c αi = αj − δij αi 2 wi wj wi c = αi (αj − δij ) , wi wj

(3.32) (3.33) (3.34) (3.35)

where δij (again) denotes Kronecker’s delta (2.95). Alternative, the second derivatives of the Cobb-Douglas cost function with respect to the input prices can be written as: ∂2c fi fj fi = − δij , ∂wi ∂wj c wi

(3.36)

where fi = ∂c/∂wi indicates the first derivative. We start with checking concavity in input prices of the Cobb-Douglas cost function without homogeneity imposed. As argued in section 2.4.11.1, we do the calculations with the predicted dependent variables rather than with the observed dependent variables.2 We can use following command to obtain the total costs which are predicted by the Cobb-Douglas cost function without homogeneity imposed: > dat$costCD cCap cLab cMat dat$hCapCap dat$hLabLab dat$hMatMat dat$hCapLab dat$hCapMat dat$hLabMat hessian hessian[ 1, 1 ] hessian[ 2, 2 ] hessian[ 3, 3 ] hessian[ 1, 2 ] det( hessian[-1,-1] ) [1] 42508376 > det( hessian ) [1] 121651514835 As all three second-order principal minors are non-negative, the conditions regarding the secondorder principal minors are all fulfilled. However, the third-order principal minor is strictly positive, which indicates that this Hessian matrix is not negative semidefinite and consequently, the Cobb-Douglas cost function is not concave at the first observation.3 3

Please note that this Hessian matrix is not positive semidefinite either, because the first-order principal minors of this Hessian are negative, while positive semidefiniteness requires that all principal minors are non-negative. Hence, the Cobb-Douglas cost function is neither concave nor convex at the first observation.

175

3 Dual Approach: Cost Functions We can check the semidefiniteness of a matrix more conveniently with the command semidefiniteness (package miscTools), which (by default) checks the signs of the principal minors and returns a logical value indicating whether the necessary and sufficient conditions for negative or positive semidefiniteness are fulfilled: > semidefiniteness( hessian, positive = FALSE ) [1] FALSE In the following, we will check whether concavity in input prices is fulfilled at each observation in the sample by stacking the Hessian matrices of all observations upon each other to a threedimensional array: > hessianCD hessianCD[ 1, 1, ] hessianCD[ 2, 2, ] hessianCD[ 3, 3, ] hessianCD[ 1, 2, ] semidefiniteness( hessianHom, positive = FALSE ) [1] TRUE In the following, we will check whether concavity in input prices is fulfilled at each observation in the sample: > hessianCDHom hessianCDHom[ 1, 1, ] hessianCDHom[ 2, 2, ] hessianCDHom[ 3, 3, ] hessianCDHom[ 1, 2, ] compPlot( chMat * dat$costCDHom / dat$pMat, dat$qMat ) > compPlot( chCap * dat$costCDHom / dat$pCap, dat$qCap, log = "xy" ) > compPlot( chLab * dat$costCDHom / dat$pLab, dat$qLab, log = "xy" ) > compPlot( chMat * dat$costCDHom / dat$pMat, dat$qMat, log = "xy" ) The resulting graphs are shown in figure 3.1. These results confirm results based on the production function: most firms should increase the use of materials and decrease the use of capital goods.

3.2.8 Optimal cost shares Given Shepard’s Lemma, the optimal cost shares derived from a Cobb-Douglas cost function are equal to the coefficients of the (logarithmic) input prices: αi =

∂c(w, y) wi wi wi xi (w, y) ∂ ln c(w, y) = = xi (w, y) = = si (w, y), ∂ ln wi ∂wi ∂c(w, y) c(w, y) c(w, y)

where si = wi xi /c are the cost shares.

179

(3.41)

100000

0

20000

60000

qMat observed

800000 400000

qLab observed

4e+05 2e+05 0e+00

qCap observed

1200000

3 Dual Approach: Cost Functions

0e+00

2e+05

4e+05

0

800000

1200000

20000

qLab optimal

60000

100000

qMat optimal

5e+03

2e+04

1e+05

qCap optimal

5e+05

5e+04 5e+03

2e+04

qMat observed

5e+05 5e+04

2e+05

qLab observed

1e+05 2e+04 5e+03

qCap observed

5e+05

qCap optimal

400000

5e+04

2e+05

5e+05

5e+03

qLab optimal

Figure 3.1: Observed and optimal input quantities

180

2e+04

5e+04

qMat optimal

3 Dual Approach: Cost Functions The following commands draw histograms of the observed cost shares and compare them to the optimal cost shares, which are predicted by our Cobb-Douglas cost function with linear homogeneity imposed: > hist( dat$pCap * dat$qCap / dat$cost ) > abline( v = chCap, lwd = 3

)

> hist( dat$pLab * dat$qLab / dat$cost ) > abline( v = chLab, lwd = 3

)

> hist( dat$pMat * dat$qMat / dat$cost )

0.0

0.1

0.2

0.3

0.4

25 15 0

5

10

Frequency

20 15 0

0

5

5

10

Frequency

20

25

35 30 25 20 15 10

Frequency

30

) 30

> abline( v = chMat, lwd = 3

0.3

cost share Cap

0.4

0.5

0.6

0.7

0.8

0.1

0.2

cost share Lab

0.3

0.4

0.5

0.6

cost share Mat

Figure 3.2: Observed and optimal costs shares The resulting graphs are shown in figure 3.2. These results confirm earlier results: most firms should increase the use of materials and decrease the use of capital goods.

3.2.9 Derived input demand functions Shepard’s Lemma says that the partial derivatives of the cost functions with respect to the input prices are the conditional input demand functions. Therefore, the input demand functions based on a Cobb-Douglas cost function are equal to the right-hand side of equation (3.31): xi (w, y) =

∂c(w, y) c(w, y) = αi ∂wi wi

(3.42)

Input demand functions should be homogeneous of degree zero in input prices: xi (t w, y) = xi (w, y)

181

(3.43)

3 Dual Approach: Cost Functions This condition is fulfilled for the input demand functions derived from any linearly homogeneous Cobb-Douglas cost function: xi (t w, y) = αi

t c(w, y) c(w, y) c(t w, y) = αi = αi = xi (w, y) t wi t wi wi

(3.44)

Furthermore, input demand functions should be symmetric with respect to input prices: ∂xi (t w, y) ∂xj (t w, y) = ∂wj ∂wi

(3.45)

This condition is fulfilled for the input demand functions derived from any Cobb-Douglas cost function: ∂xi (w, y) αi ∂c(w, y) αi c(w, y) αi αj = = αj = c(w, y) ∀ i 6= j ∂wj wi ∂wj wi wj wi wj ∂xj (w, y) αj ∂c(w, y) αj c(w, y) αi αj = = αi = c(w, y) ∀ i 6= j ∂wi wj ∂wi wj wi wi wj

(3.46) (3.47)

Finally, input demand functions should fulfill the negativity condition: ∂xi (t w, y) ≤0 ∂wi

(3.48)

This condition is fulfilled for the input demand functions derived from any linearly homogeneous Cobb-Douglas cost function that is monotonically increasing in all input prices (as this implies 0 ≤ αi ≤ 1): αi ∂c(w, y) c(w, y) ∂xi (w, y) = − αi ∂wi wi ∂wi wi2 αi c(w, y) c(w, y) = αi − αi wi wi wi2 c(w, y) = αi (αi − 1) ≤ 0 wi2

(3.49) (3.50) (3.51)

3.2.10 Derived input demand elasticities Based on the derived input demand functions (3.42), we can derive the conditional input demand elasticities: ∂xi (w, y) wj ∂wj xi (w, y) wj c(w, y) wj αi ∂c(w, y) = − δij αi wi ∂wj xi (w, y) wj2 xi (w, y)

ij (w, y) =

=

αi c(w, y) wj c(w, y) αj − δij αi wi wj xi (w, y) wi xi (w, y)

182

(3.52) (3.53) (3.54)

3 Dual Approach: Cost Functions c(w, y) αi − δij wi xi (w, y) si (w, y) αi αj αi = − δij si (w, y) si (w, y) = αi αj

= αj − δij

(3.55) (3.56) (3.57)

y ∂xi (w, y) ∂y xi (w, y) y ∂c(w, y) αi = ∂y wi xi (w, y) c(w, y) αi y = αy y wi xi (w, y) y c(w, y) = αi αy y wi xi (w, y) c(w, y) = αi αy wi xi (w, y) αi = αy si (w, y)

iy (w, y) =

= αy

(3.58) (3.59) (3.60) (3.61) (3.62) (3.63) (3.64)

All derived input demand elasticities based on our estimated Cobb-Douglas cost function with linear homogeneity imposed are presented in table 3.1. If the price of capital increases by one percent, the cost-minimizing firm will decrease the use of capital by 0.93% and increase the use of labor and materials by 0.07% each. If the price of labor increases by one percent, the cost-minimizing firm will decrease the use of labor by 0.55% and increase the use of capital and materials by 0.45% each. If the price of materials increases by one percent, the cost-minimizing firm will decrease the use of materials by 0.52% and increase the use of capital and labor by 0.48% each. If the cost-minimizing firm increases the output quantity by one percent, (s)he will increase all input quantities by 0.37%. The price elasticities derived from the Cobb-Douglas cost function with linear homogeneity imposed are rather similar to the price elasticities derived from the Cobb-Douglas production function but the elasticities with respect to the output quantity are rather dissimilar (compare Tables 2.1 and 3.1). In theory, elasticities derived from a cost function, which corresponds to a specific production function, should be identical to elasticities which are directly derived from the production function. However, although our production function and cost function are supposed to model the same production technology, their elasticities are not the same. These differences arise from different econometric assumptions (e.g. exogeneity of explanatory variables) and the disturbance terms, which differ between both models so that the production technology is fitted differently. Given Euler’s theorem and the cost function’s homogeneity in input prices, following condition

183

3 Dual Approach: Cost Functions

Table 3.1: Conditional demand elasticities derived from Cobb-Douglas cost function (with linear homogeneity imposed) wcap wlab wmat y xcap -0.93 0.45 0.48 0.37 xlab 0.07 -0.55 0.48 0.37 xmat 0.07 0.45 -0.52 0.37 for the price elasticities can be obtained: X

ij = 0 ∀ i

(3.65)

j

The input demand elasticities derived from any linearly homogeneous Cobb-Douglas cost function fulfill the homogeneity condition: X

ij (w, y) =

j

X

(αj − δij ) =

j

X

αj −

X

j

δij = 1 − 1 = 0 ∀ i

(3.66)

j

As we computed the elasticities in table 3.1 based on the Cobb-Douglas function with linear homogeneity imposed, these conditions are fulfilled for these elasticities, e.g. for i = cap: -0.93 + 0.45 + 0.48 = 0. It follows from the necessary conditions for the concavity of the cost function that all own-price elasticities are non-positive: ii ≤ 0 ∀ i

(3.67)

The input demand elasticities derived from any linearly homogeneous Cobb-Douglas cost function that is monotonically increasing in all input prices fulfill the negativity condition, because linear P

homogeneity (

i αi

= 1) and monotonicity (αi ≥ 0 ∀ i) together imply that all αi s (optimal cost

shares) are between zero and one (0 ≤ αi ≤ 1 ∀ i): ii = αi − 1 ≤ 0 ∀ i

(3.68)

As our Cobb-Douglas function with linear homogeneity imposed fulfills the homogeneity, monotonicity, and concavity condition, the own-price elasticities (on the diagonal) in table 3.1 fulfill the negativity conditions. The symmetry condition for derived demand elasticities si ij = sj ji ∀ i, j

(3.69)

is fulfilled for any Cobb-Douglas cost function: si ij = αi αj = αj αi = sj ji ∀ i 6= j

(3.70)

si ij = si ii = sj jj = sj ji ∀ i = j

(3.71)

184

3 Dual Approach: Cost Functions Hence, the symmetry condition is also fulfilled for the elasticities in table 3.1, e.g. scap cap,lab = αcap cap,lab = 0.07 · 0.45 is equal to slab lab,cap = αlab lab,cap = 0.45 · 0.07.

3.2.11 Cost flexibility and elasticity of size The coefficient of the (logarithmic) output quantity is equal to the “cost flexibility” (3.3). A value of 0.37 (as in our estimated Cobb-Douglas cost function with linear homogeneity in input prices imposed) means that a 1% increase in the output quantity results in a cost increase of 0.37%. The “elasticity of size” is the inverse of the cost flexibility (3.4). A value of 2.67 (as derived from our estimated Cobb-Douglas cost function with linear homogeneity in input prices imposed) means that if minimum costs are increased by 1%, the output quantity increases by 2.67%.

3.2.12 Marginal costs, average costs, and total costs Marginal costs can be calculated by ∂c(w, y) c(w, y) = αy ∂y y

(3.72)

These marginal costs should be linearly homogeneous in input prices: ∂c(w, y) ∂c(t w, y) =t ∂y ∂y

(3.73)

This condition is fulfilled for the marginal costs derived from a linearly homogeneous CobbDouglas cost function: ∂c(t w, y) ∂c(w, y) c(t w, y) t c(w, y) c(w, y) = αy = αy = t αy =t ∂y y y y ∂y

(3.74)

We can compute the marginal costs by following command: > chOut dat$margCost hist( dat$margCost, 20 ) The resulting graph is shown in figure 3.3. It indicates that producing one additional output unit increases the costs of most firms by around 0.08 monetary units. Furthermore, we can check if the marginal costs are equal to the output prices, which is a first-order condition for profit maximization: > compPlot( dat$pOut, dat$margCost ) > compPlot( dat$pOut, dat$margCost, log = "xy" )

185

0 5

15

Frequency

25

3 Dual Approach: Cost Functions

0.0

0.1

0.2

0.3

0.4

0.5

margCost

0.50 0.10 0.02

margCost

2.00

0.0 0.5 1.0 1.5 2.0 2.5 3.0

margCost

Figure 3.3: Marginal costs

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.02

0.10

pOut

0.50 pOut

Figure 3.4: Marginal costs and output prices

186

2.00

3 Dual Approach: Cost Functions The resulting graphs are shown in figure 3.4. The marginal costs of all firms are considerably smaller than their output prices. Hence, all firms would gain from increasing their output level. This is not surprising for a technology with large economies of scale. Now, we analyze, how the marginal costs depend on the output quantity: > plot( dat$qOut, dat$margCost )

0.0e+00

0.20 0.05

0.10

margCost

0.3 0.2 0.1

margCost

0.4

0.5

0.50

> plot( dat$qOut, dat$margCost, log = "xy" )

1.0e+07

2.0e+07

1e+05

5e+05

qOut

5e+06 qOut

Figure 3.5: Marginal costs depending on output quantity and firm size The resulting graphs are shown in figure 3.5. Due to the large economies of size, the marginal costs are decreasing with the output quantity. The relation between output quantity and marginal costs in a Cobb-Douglas cost function can be analyzed by taking the first derivative of the marginal costs (3.72) with respect to the output quantity: 

∂M C = ∂y

∂ αy c(w,y) y



∂y αy ∂c(w, y) c(w, y) = − αy y ∂y y2 αy c(w, y) c(w, y) = αy − αy y y y2 c = αy 2 (αy − 1) y

(3.75) (3.76) (3.77) (3.78)

As αy , c, and y 2 should always be positive, the marginal costs are (globally) increasing in the output quantity, if there are decreasing returns to size (i.e. αy > 1) and the marginal costs are (globally) decreasing in the output quantity, if there are increasing returns to size (i.e. αy < 1). Now, we illustrate our estimated model by drawing the total cost curve for output quantities between the minimum and the maximum output level in the sample, where we use the sample

187

3 Dual Approach: Cost Functions means of the input prices. Furthermore, we draw the average cost curve and the marginal cost curve for the above-mentioned output quantities and input prices: > y chInt costs plot( y, costs, type = "l" ) > # average costs > plot( y, costs/y, type = "l" ) > # marginal costs > lines( y, chOut * costs / y, lty = 2 ) > legend( "right", lty = c( 1, 2 ),

0.0e+00

1.0e+07

2.0e+07

1.5 1.0 0.5

average costs marginal costs

0.0

200000

600000

1000000

average costs, marginal costs

legend = c( "average costs", "marginal costs" ) )

total costs

+

0.0e+00

y

1.0e+07

2.0e+07

y

Figure 3.6: Total, marginal, and average costs The resulting graphs are shown in figure 3.6. As the marginal costs are equal to the average costs multiplied by a fixed factor, αy (see equation 3.72), the average cost curve and the marginal cost curve of a Cobb-Douglas cost function cannot intersect. Hence, the average costs curve based on a Cobb-Douglas cost function does not have a minimum (for finite output quantities y).

188

3 Dual Approach: Cost Functions

3.3 Cobb-Douglas short-run cost function 3.3.1 Specification Given the general specification of a short-run cost function (3.9), a Cobb-Douglas short-run cost function is



cv = A 

 Y

wiαi  

i∈N 1

 Y

α xj j  y αy ,

(3.79)

j∈N 2

where cv denotes the variable costs as defined in (1.3), N 1 is a vector of the indices of the variable inputs, and N 2 is a vector of the indices of the quasi-fixed inputs. The Cobb-Douglas short-run cost function can be linearized to ln cv = α0 +

X

αi ln wi +

i∈N 1

X

αj ln xj + αy ln y

(3.80)

j∈N 2

with α0 = ln A.

3.3.2 Estimation The following commands estimate a Cobb-Douglas short-run cost function with capital as a quasi-fixed input and summarize the results: > costCDSR summary( costCDSR ) Call: lm(formula = log(vCost) ~ log(pLab) + log(pMat) + log(qCap) + log(qOut), data = dat) Residuals: Min

1Q

Median

3Q

Max

-0.73935 -0.20934 -0.00571

0.20729

0.71633

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

5.66013

0.42523

13.311

< 2e-16 ***

log(pLab)

0.45683

0.13819

3.306

0.00121 **

log(pMat)

0.44144

0.07715

5.722 6.50e-08 ***

log(qCap)

0.19174

0.04034

4.754 5.05e-06 ***

log(qOut)

0.29127

0.03318

8.778 6.49e-15 ***

---

189

3 Dual Approach: Cost Functions Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3183 on 135 degrees of freedom Multiple R-squared:

0.7265,

Adjusted R-squared:

F-statistic: 89.63 on 4 and 135 DF,

0.7183

p-value: < 2.2e-16

3.3.3 Properties This short-run cost function is (significantly) increasing in the prices of the variable inputs (labor and materials) as the coefficient of the labor price (0.457) and the coefficient of the materials price (0.441) are both positive. However, this short-run cost function is not linearly homogeneous in input prices, as the coefficient of the labor price and the coefficient of the materials price do not sum up to one (0.457 + 0.441 = 0.898). The short-run cost function is increasing in the output quantity with a short-run cost flexibility of 0.291, which corresponds to a short-run elasticity of size of 3.433. However, this short-run cost function is increasing in the quantity of the fixed input (capital), as the corresponding coefficient is (significantly) positive (0.192) which contradicts microeconomic theory. This would mean that the apple producers could reduce variable costs (costs from labor and materials) by reducing the capital input (e.g. by destroying their apple trees and machinery), while still producing the same amount of apples. Producing the same output level with less of all inputs is not plausible. The violation of the monotonicity property regarding the capital input is probably caused by model misspecification. For instance, the assumption that the capital input quantity is exogenously given may not hold; the capital input quantity could be positively correlated with the error term, e.g. if poorly performing firms (with higher variable costs) use more capital than well performing firms (with lower variable costs) to produce the same output quantity.

3.3.4 Estimation with linear homogeneity in input prices imposed We can impose linear homogeneity in the prices of the variable inputs as we did with the (longrun) cost function (see equations 3.24 to 3.28): ln

X X c wi = α0 + αi ln + αj ln xj + αy ln y wk wk 1 2 i∈N \k

(3.81)

j∈N

with k ∈ N 1 . We can estimate a Cobb-Douglas short-run cost function with capital as a quasifixed input and linear homogeneity in input prices imposed by the command: > costCDSRHom summary( costCDSRHom ) Call: lm(formula = log(vCost/pMat) ~ log(pLab/pMat) + log(qCap) + log(qOut),

190

3 Dual Approach: Cost Functions data = dat) Residuals: Min

1Q

Median

3Q

Max

-0.78305 -0.20539 -0.00265

0.19533

0.71792

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

5.67882

0.42335

13.414

< 2e-16 ***

log(pLab/pMat)

0.53487

0.06781

7.888 9.00e-13 ***

log(qCap)

0.18774

0.03978

4.720 5.79e-06 ***

log(qOut)

0.29010

0.03306

8.775 6.33e-15 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3176 on 136 degrees of freedom Multiple R-squared:

0.5963,

F-statistic: 66.97 on 3 and 136 DF,

Adjusted R-squared:

0.5874

p-value: < 2.2e-16

We can obtain the coefficient of the materials price from the homogeneity condition (3.24): 1 − 0.535 = 0.465. We can test the homogeneity restriction by a likelihood ratio test: > lrtest( costCDSRHom, costCDSR ) Likelihood ratio test Model 1: log(vCost/pMat) ~ log(pLab/pMat) + log(qCap) + log(qOut) Model 2: log(vCost) ~ log(pLab) + log(pMat) + log(qCap) + log(qOut) #Df

LogLik Df

1

5 -36.055

2

6 -35.838

Chisq Pr(>Chisq)

1 0.4356

0.5093

Given the large P -value, we can conclude that the data do not contradict the linear homogeneity in the prices of the variable inputs. While the linear homogeneity in the prices of all variable inputs is accepted and the short-run cost function is still increasing in the output quantity and the prices of all variable inputs, the estimated short-run cost function is still increasing in the capital quantity, which contradicts microeconomic theory. Therefore, a further microeconomic analysis with this function is not reasonable.

191

3 Dual Approach: Cost Functions

3.4 Cobb-Douglas cost function with multiple outputs 3.4.1 Specification The Cobb-Douglas cost function with multiple outputs is a minor extension of the Cobb-Douglas cost function with a single output as defined in equations (3.15) and (3.16): N Y

c=A

M Y

!

wiαi

! βm ym

(3.82)

m=1

i=1

This function can be linearized to: ln c = α0 +

N X

M X

αi ln wi +

βm ln ym

(3.83)

m=1

i=1

with α0 = ln A.

3.4.2 Properties The Cobb-Douglas multiple-output cost function is never quasi-convex in output quantities because it always has convex iso-cost curves for different output-output combinations.

3.4.3 Cost flexibility and elasticity of size The cost flexibility is: M X

βm

(3.84)

j=1

and the elasticity of size is: 

∗ = 

M X

−1

βm 

.

(3.85)

j=1

3.5 Translog cost function 3.5.1 Specification The general specification of a Translog cost function is ln c(w, y) = α0 + +

+

1 2

N X

αi ln wi i=1 N X N X

+ αy ln y

1 αij ln wi ln wj + αyy (ln y)2 2 i=1 j=1

N X

αiy ln wi ln y

i=1

192

(3.86)

3 Dual Approach: Cost Functions with αij = αji ∀ i, j.

3.5.2 Estimation The Translog cost function can be estimated by following command: > costTL summary( costTL ) Call: lm(formula = log(cost) ~ log(pCap) + log(pLab) + log(pMat) + log(qOut) + I(0.5 * log(pCap)^2) + I(0.5 * log(pLab)^2) + I(0.5 * log(pMat)^2) + I(log(pCap) * log(pLab)) + I(log(pCap) * log(pMat)) + I(log(pLab) * log(pMat)) + I(0.5 * log(qOut)^2) + I(log(pCap) * log(qOut)) + I(log(pLab) * log(qOut)) + I(log(pMat) * log(qOut)), data = dat) Residuals: Min

1Q

Median

3Q

Max

-0.73251 -0.18718

0.02001

0.15447

0.82858

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

25.383429

3.511353

7.229 4.26e-11 ***

log(pCap)

0.198813

0.537885

0.370 0.712291

log(pLab)

-0.024792

2.232126

-0.011 0.991156

log(pMat)

-1.244914

1.201129

-1.036 0.301992

log(qOut)

-2.040079

0.510905

-3.993 0.000111 ***

I(0.5 * log(pCap)^2)

-0.095173

0.105158

-0.905 0.367182

I(0.5 * log(pLab)^2)

-0.503168

0.943390

-0.533 0.594730

I(0.5 * log(pMat)^2)

0.529021

0.337680

1.567 0.119728

I(log(pCap) * log(pLab)) -0.746199

0.244445

I(log(pCap) * log(pMat))

0.182268

0.130463

1.397 0.164865

I(log(pLab) * log(pMat))

0.139429

0.433408

0.322 0.748215

I(0.5 * log(qOut)^2)

0.164075

0.041078

3.994 0.000110 ***

193

-3.053 0.002772 **

3 Dual Approach: Cost Functions I(log(pCap) * log(qOut)) -0.028090

0.042844

-0.656 0.513259

I(log(pLab) * log(qOut))

0.007533

0.171134

0.044 0.964959

I(log(pMat) * log(qOut))

0.048794

0.092266

0.529 0.597849

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3043 on 125 degrees of freedom Multiple R-squared:

0.7682,

Adjusted R-squared:

F-statistic: 29.59 on 14 and 125 DF,

0.7423

p-value: < 2.2e-16

As the Cobb-Douglas cost function is nested in the Translog cost function, we can use a statistical test to check whether the Cobb-Douglas cost function fits the data as good as the Translog cost function: > lrtest( costCD, costTL ) Likelihood ratio test Model 1: log(cost) ~ log(pCap) + log(pLab) + log(pMat) + log(qOut) Model 2: log(cost) ~ log(pCap) + log(pLab) + log(pMat) + log(qOut) + I(0.5 * log(pCap)^2) + I(0.5 * log(pLab)^2) + I(0.5 * log(pMat)^2) + I(log(pCap) * log(pLab)) + I(log(pCap) * log(pMat)) + I(log(pLab) * log(pMat)) + I(0.5 * log(qOut)^2) + I(log(pCap) * log(qOut)) + I(log(pLab) * log(qOut)) + I(log(pMat) * log(qOut)) #Df 1 2

LogLik Df

Chisq Pr(>Chisq)

6 -44.867 16 -24.149 10 41.435

9.448e-06 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Given the very small P -value, we can conclude that the Cobb-Douglas cost function is not suitable for analyzing the production technology in our data set.

3.5.3 Linear homogeneity in input prices Linear homogeneity of a Translog cost function requires ln(t c(w, y)) = ln c(t w, y) ln t + ln c(w, y) = α0 + +

1 2

N X

αi ln(t i=1 N X N X

(3.87) wi ) + αy ln y

1 αij ln(t wi ) ln(t wj ) + αyy (ln y)2 2 i=1 j=1

194

(3.88)

3 Dual Approach: Cost Functions

+

N X

αiy ln(t wi ) ln y

i=1

= α0 + 1 + 2 +

N X

αi ln(t) i=1 N X N X

+

N X

αi ln(wi ) + αy ln y

(3.89)

i=1

N X N 1X αij ln(t) ln(t) + αij ln(t) ln(wj ) 2 i=1 j=1 i=1 j=1

N X N X N N 1X 1X αij ln(wi ) ln(t) + αij ln(wi ) ln(wj ) 2 i=1 j=1 2 i=1 j=1

N N X X 1 2 αiy ln(t) ln y + αiy ln(wi ) ln y + αyy (ln y) + 2 i=1 i=1

= α0 + ln(t)

N X i=1

αi +

N X

αi ln(wi ) + αy ln y

(3.90)

i=1 N X N X

+

N N X X 1 1 ln(t) ln(t) ln(wj ) αij αij + ln(t) 2 2 i=1 j=1 j=1 i=1

+

N N N X N X X 1X 1 αij + ln(wi ) αij ln(wi ) ln(wj ) ln(t) 2 2 i=1 j=1 j=1 i=1

N N X X 1 2 + αyy (ln y) + ln(t) ln y αiy ln(wi ) ln y αiy + 2 i=1 i=1

= ln c(w, y) + ln(t)

N X

αi

(3.91)

i=1 N N X X

+

N N X X 1 1 αij + ln(t) αij ln(wj ) ln(t) ln(t) 2 2 i=1 i=1 j=1 j=1

+

N N N X X X 1 ln(t) ln(wi ) αij + ln(t) ln y αiy 2 i=1 j=1 i=1

N X N N N X X X 1 1 ln t = ln(t) αi + ln(t) ln(t) αij + ln(t) ln(wj ) αij 2 2 i=1 i=1 j=1 j=1 i=1 N X

+

N N N X X X 1 αij + ln(t) ln y αiy ln(wi ) ln(t) 2 j=1 i=1 i=1

N X

N X N N N X X 1 1X 1= αi + ln(t) αij + ln(wj ) αij 2 2 j=1 i=1 i=1 j=1 i=1

+

(3.92)

N N N X X 1X ln(wi ) αij + ln y αiy 2 i=1 j=1 i=1

195

(3.93)

3 Dual Approach: Cost Functions Hence, the homogeneity condition is only globally fulfilled (i.e. no matter which values t, w, and y have) if the following parameter restrictions hold: N X i=1 N X

αi = 1

(3.94) αij =αji

αij = 0 ∀ j ←−−−−→

i=1 N X

N X

αij = 0 ∀ i

(3.95)

j=1

αiy = 0

(3.96)

i=1

We can see from the estimates above that these conditions are not fulfilled in our Translog cost function. For instance, according to condition (3.94), the first-order coefficients of the input prices should sum up to one but our estimates sum up to 0.199 + (−0.025) + (−1.245) = −1.071. Hence, the homogeneity condition is not fulfilled in our estimated Translog cost function.

3.5.4 Estimation with linear homogeneity in input prices imposed In order to impose linear homogeneity in input prices, we can rearrange these restrictions to get αN = 1 − αN j = − αiN = −

N −1 X

αi

i=1 N −1 X i=1 N −1 X

(3.97)

αij ∀ j

(3.98)

αij ∀ i

(3.99)

j=1

αN y = −

N −1 X

αiy

(3.100)

i=1

Replacing αN , αN y and all αiN and αjN in equation (3.86) by the right-hand sides of equations (3.97) to (3.100) and re-arranging, we get N −1 X c(w, y) wi ln = α0 + αi ln + αy ln y wN w N i=1

+

+

(3.101)

−1 N −1 X 1 NX wi wj 1 αij ln ln + αyy (ln y)2 2 j=1 i=1 wN wN 2 N −1 X i=1

αiy ln

wi ln y. wN

This Translog cost function with linear homogeneity imposed can be estimated by following command:

196

3 Dual Approach: Cost Functions > costTLHom summary( costTLHom ) Call: lm(formula = log(cost/pMat) ~ log(pCap/pMat) + log(pLab/pMat) + log(qOut) + I(0.5 * log(pCap/pMat)^2) + I(0.5 * log(pLab/pMat)^2) + I(log(pCap/pMat) * log(pLab/pMat)) + I(0.5 * log(qOut)^2) + I(log(pCap/pMat) * log(qOut)) + I(log(pLab/pMat) * log(qOut)), data = dat) Residuals: Min

1Q

Median

3Q

Max

-0.6860 -0.2086

0.0192

0.1978

0.8281

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

23.714976

3.445289

6.883 2.24e-10 ***

log(pCap/pMat)

0.306159

0.525789

0.582 0.561383

log(pLab/pMat)

1.093860

1.169160

0.936 0.351216

-1.933605

0.501090

I(0.5 * log(pCap/pMat)^2)

0.025951

0.089977

0.288 0.773486

I(0.5 * log(pLab/pMat)^2)

0.716467

0.338049

2.119 0.035957 *

I(log(pCap/pMat) * log(pLab/pMat)) -0.292889

0.142710

-2.052 0.042144 *

I(0.5 * log(qOut)^2)

0.158662

0.039866

I(log(pCap/pMat) * log(qOut))

-0.048274

0.040025

-1.206 0.229964

I(log(pLab/pMat) * log(qOut))

0.008363

0.096490

0.087 0.931067

log(qOut)

-3.859 0.000179 ***

3.980 0.000114 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3089 on 130 degrees of freedom Multiple R-squared:

0.6377,

F-statistic: 25.43 on 9 and 130 DF,

Adjusted R-squared:

0.6126

p-value: < 2.2e-16

We can use a likelihood ratio test to compare this function with the unconstrained Translog cost function (3.86):

197

3 Dual Approach: Cost Functions > lrtest( costTL, costTLHom ) Likelihood ratio test Model 1: log(cost) ~ log(pCap) + log(pLab) + log(pMat) + log(qOut) + I(0.5 * log(pCap)^2) + I(0.5 * log(pLab)^2) + I(0.5 * log(pMat)^2) + I(log(pCap) * log(pLab)) + I(log(pCap) * log(pMat)) + I(log(pLab) * log(pMat)) + I(0.5 * log(qOut)^2) + I(log(pCap) * log(qOut)) + I(log(pLab) * log(qOut)) + I(log(pMat) * log(qOut)) Model 2: log(cost/pMat) ~ log(pCap/pMat) + log(pLab/pMat) + log(qOut) + I(0.5 * log(pCap/pMat)^2) + I(0.5 * log(pLab/pMat)^2) + I(log(pCap/pMat) * log(pLab/pMat)) + I(0.5 * log(qOut)^2) + I(log(pCap/pMat) * log(qOut)) + I(log(pLab/pMat) * log(qOut)) #Df

LogLik Df

Chisq Pr(>Chisq)

1

16 -24.149

2

11 -29.014 -5 9.7309

0.08323 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The null hypothesis, linear homogeneity in input prices, is rejected at the 10% significance level but not at the 5% level. Given the importance of microeconomic consistency and that 5% is the standard significance level, we continue our analysis with the Translog cost function with linear homogeneity in input prices imposed. Furthermore, we can use a likelihood ratio test to compare this function with the Cobb-Douglas cost function with homogeneity imposed (3.28): > lrtest( costCDHom, costTLHom ) Likelihood ratio test Model 1: log(cost/pMat) ~ log(pCap/pMat) + log(pLab/pMat) + log(qOut) Model 2: log(cost/pMat) ~ log(pCap/pMat) + log(pLab/pMat) + log(qOut) + I(0.5 * log(pCap/pMat)^2) + I(0.5 * log(pLab/pMat)^2) + I(log(pCap/pMat) * log(pLab/pMat)) + I(0.5 * log(qOut)^2) + I(log(pCap/pMat) * log(qOut)) + I(log(pLab/pMat) * log(qOut)) #Df

LogLik Df

1

5 -44.878

2

11 -29.014

Chisq Pr(>Chisq)

6 31.727

1.84e-05 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

198

3 Dual Approach: Cost Functions Again, the Cobb-Douglas functional form is clearly rejected by the data in favor of the Translog functional form. We can compute the total costs that are predicted by our estimated Translog cost function and compare them to the observed costs by following commands: > dat$costTLHom compPlot( dat$cost, dat$costTLHom )

2e+06 2e+05

5e+05

dat$costTLHom

2500000 1000000 0

dat$costTLHom

> compPlot( dat$cost, dat$costTLHom, log = "xy" )

0

1000000

2500000

2e+05

dat$cost

5e+05

2e+06

dat$cost

Figure 3.7: Translog cost function: observed vs. predicted costs The resulting graphs are presented in figure 3.7. Some parameters of the Translog cost function with linear homogeneity imposed (3.101) have not been directly estimated (αN , αN y , all αiN , all αjN ) but they can be retrieved from the (directly) estimated parameters and equations (3.97) to (3.100). Please note that the specification in equation (3.101) is used for the econometric estimation only; after retrieving the non-estimated parameters, we can do our analysis based on equation (3.86). To facilitate the further analysis, we create short-cuts of all estimated parameters and obtain the parameters that have not been directly estimated: > ch0 ch1 ch2 ch3 chy ch11 ch22 chyy ch12 matrix( c( ch11, ch12, ch13, ch21, ch22, ch23, ch31, ch32, ch33 ), ncol=3 ) [,1] [1,]

[,2]

[,3]

0.02595083 -0.2928892

0.2669384

[2,] -0.29288920 [3,]

0.7164670 -0.4235778

0.26693837 -0.4235778

0.1566394

> # alpha_iy, alpha_yy > unname( c( ch1y, ch2y, ch3y, chyy ) ) [1] -0.048274484

0.008362717

0.039911768

0.158661757

3.5.5 Cost flexibility and elasticity of size Based on the estimated parameters, we can calculate the cost flexibilities and the elasticities of size. The cost flexibilities derived from a Translog cost function (3.86) are N X ∂ ln c(w, y) = αy + αiy ln wi + αyy ln y ∂ ln y i=1

(3.102)

and the elasticities of size are—as always—their inverses: ∂ ln y = ∂ ln c



∂ ln c(w, y) ∂ ln y

−1

(3.103)

We can calculate the cost flexibilities and the elasticities of size with following commands:

200

3 Dual Approach: Cost Functions > dat$costFlex dat$elaSize hist( dat$costFlex ) > hist( dat$elaSize )

0.2

0.4

0.6

0.8

cost flexibility

50 40 30 10 0

20 0

5 0

0.0

20

Frequency

80 60 40

Frequency

20 15 10

Frequency

25

30

120

60

35

> hist( dat$elaSize[ dat$elaSize > 0 & dat$elaSize < 10 ] )

−20

0

20

40

60

80

100

elasticity of size

2

4

6

8

10

elasticity of size

Figure 3.8: Translog cost function: cost flexibility and elasticity of size The resulting graphs are presented in figure 3.8. Only 1 out of 140 cost flexibilities is negative. Hence, the estimated Translog cost function is to a very large extent increasing in the output quantity. All cost flexibilities are lower than one, which indicates that all apple producers operate under increasing returns to size. Most cost flexibilities are around 0.5, which corresponds to an elasticity of size of 2. Hence, if the apple producers increase their output quantity by one percent, the total costs of most producers increases by around 0.5 percent. Or—the other way round—if the apple producers increase their input use so that their costs increase by one percent, the output quantity of most producers would increase by around two percent. With the following commands, we visualize the relationship between output quantity and elasticity of size > plot( dat$qOut, dat$elaSize ) > abline( 1, 0 ) > plot( dat$qOut, dat$elaSize, ylim = c( 0, 10 ) ) > abline( 1, 0 ) > plot( dat$qOut, dat$elaSize, ylim = c( 0, 10 ), log = "x" ) > abline( 1, 0 ) The resulting graphs are shown in figure 3.9. With increasing output quantity, the elasticity of size approaches one (from above). Hence, small apple producers could gain a lot from increasing

201

10 8 6 2

4

dat$elaSize

8

0.0e+00

1.0e+07

2.0e+07

0

0

−20

0

2

4

6

dat$elaSize

60 40 20

dat$elaSize

80

10

100

3 Dual Approach: Cost Functions

0.0e+00

1.0e+07

qOut

2.0e+07

1e+05

qOut

5e+05

5e+06 qOut

Figure 3.9: Translog cost function: output quantity and elasticity of size their size, while large apple producers would gain much less from increasing their size. However, even the largest producers still gain from increasing their size so that the optimal firm size is larger than the largest firm in the sample.

3.5.6 Homotheticity In order to check whether the Translog cost function as specified in equation (3.86) is homothetic, we check whether we can decompose it as done in equation (3.5). This decomposition of the Translog cost function is possible if αiy = 0 ∀ i so that it can be decomposed in the following way: c0 (c1 , c2 ) = exp (α0 + c1 (w) + c2 (y)) c1 (w) =

N X

αi ln wi +

i=1

1 2

N N X X

αij ln wi ln wj

(3.104) (3.105)

i=1 j=1

1 c2 (y) = αy ln y + αyy (ln y)2 . 2

(3.106)

Hence, we can can conclude that the Translog cost function is homothetic if αiy = 0 ∀ i. The following command estimates a Translog cost function with both linear homogeneity in input prices and homotheticity imposed: > costTLHomHom summary( costTLHomHom ) Call:

202

3 Dual Approach: Cost Functions lm(formula = log(cost/pMat) ~ log(pCap/pMat) + log(pLab/pMat) + log(qOut) + I(0.5 * log(pCap/pMat)^2) + I(0.5 * log(pLab/pMat)^2) + I(log(pCap/pMat) * log(pLab/pMat)) + I(0.5 * log(qOut)^2), data = dat) Residuals: Min

1Q

Median

3Q

Max

-0.69257 -0.20604

0.00191

0.20321

0.82527

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

23.13551

3.40391

log(pCap/pMat)

-0.28784

0.20635

-1.395

log(pLab/pMat)

1.22386

0.44113

2.774

-1.91286

0.47646

I(0.5 * log(pCap/pMat)^2)

0.01609

0.08951

0.180

0.85761

I(0.5 * log(pLab/pMat)^2)

0.67114

0.28180

2.382

0.01866 *

I(log(pCap/pMat) * log(pLab/pMat)) -0.23524

0.13214

-1.780

0.07734 .

I(0.5 * log(qOut)^2)

0.03352

log(qOut)

0.16235

6.797 3.34e-10 *** 0.16539 0.00633 **

-4.015 9.93e-05 ***

4.844 3.51e-06 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3085 on 132 degrees of freedom Multiple R-squared:

0.6331,

F-statistic: 32.54 on 7 and 132 DF,

Adjusted R-squared:

0.6137

p-value: < 2.2e-16

The following two commands apply a Wald test and a likelihood-ratio test, respectively, to test the homotheticity property: > waldtest( costTLHom, costTLHomHom ) Wald test Model 1: log(cost/pMat) ~ log(pCap/pMat) + log(pLab/pMat) + log(qOut) + I(0.5 * log(pCap/pMat)^2) + I(0.5 * log(pLab/pMat)^2) + I(log(pCap/pMat) * log(pLab/pMat)) + I(0.5 * log(qOut)^2) + I(log(pCap/pMat) * log(qOut)) + I(log(pLab/pMat) * log(qOut)) Model 2: log(cost/pMat) ~ log(pCap/pMat) + log(pLab/pMat) + log(qOut) + I(0.5 * log(pCap/pMat)^2) + I(0.5 * log(pLab/pMat)^2) + I(log(pCap/pMat) * log(pLab/pMat)) + I(0.5 * log(qOut)^2) Res.Df Df

F Pr(>F)

203

3 Dual Approach: Cost Functions 1

130

2

132 -2 0.8259 0.4401

> lrtest( costTLHom, costTLHomHom ) Likelihood ratio test Model 1: log(cost/pMat) ~ log(pCap/pMat) + log(pLab/pMat) + log(qOut) + I(0.5 * log(pCap/pMat)^2) + I(0.5 * log(pLab/pMat)^2) + I(log(pCap/pMat) * log(pLab/pMat)) + I(0.5 * log(qOut)^2) + I(log(pCap/pMat) * log(qOut)) + I(log(pLab/pMat) * log(qOut)) Model 2: log(cost/pMat) ~ log(pCap/pMat) + log(pLab/pMat) + log(qOut) + I(0.5 * log(pCap/pMat)^2) + I(0.5 * log(pLab/pMat)^2) + I(log(pCap/pMat) * log(pLab/pMat)) + I(0.5 * log(qOut)^2) #Df 1 2

LogLik Df

Chisq Pr(>Chisq)

11 -29.014 9 -29.898 -2 1.7676

0.4132

None of the two tests can reject homotheticity of the estimated Translog cost funtion.

3.5.7 Marginal costs and average costs Marginal costs derived from a Translog cost function are ∂c(w, y) ∂ ln c(w, y) c(w, y) = = ∂y ∂ ln y y

αy +

N X

!

αiy ln wi + αyy ln y

i=1

c(w, y) . y

(3.107)

Hence, they are—as always—equal to the cost flexibility multiplied by total costs and divided by the output quantity. Hence, we can compute the marginal costs by: > dat$margCostTL hist( dat$margCostTL, 15 ) The resulting graph is shown in figure 3.10. It indicates that producing one additional output unit increases the costs of most firms by around 0.09 monetary units. Furthermore, we can check if the marginal costs are equal to the output prices, which is a first-order condition for profit maximization: > compPlot( dat$pOut, dat$margCostTL ) > compPlot( dat$pOut[ dat$margCostTL > 0 ], +

dat$margCostTL[ dat$margCostTL > 0 ], log = "xy" )

204

40 20 0

Frequency

3 Dual Approach: Cost Functions

−0.15

−0.05

0.05

0.15

margCostTL

2.00 0.50 0.02

0.10

margCostTL

2.0 1.0 0.0

margCostTL

3.0

Figure 3.10: Translog cost function: Marginal costs

0.0 0.5 1.0 1.5 2.0 2.5 3.0 pOut

0.02

0.10

0.50

2.00

pOut

Figure 3.11: Translog cost function: marginal costs and output prices

205

3 Dual Approach: Cost Functions The resulting graphs are shown in figure 3.11. The marginal costs of all firms are considerably smaller than their output prices. Hence, all firms would gain from increasing their output level. This is not surprising for a technology with large economies of scale. Now, we analyze, how the marginal costs depend on the output quantity: > plot( dat$qOut, dat$margCostTL )

0.0e+00

0.15 0.05 −0.15

−0.05

margCost

0.05 −0.05 −0.15

margCost

0.15

> plot( dat$qOut, dat$margCostTL, log = "x" )

1.0e+07

2.0e+07

1e+05

qOut

5e+05

5e+06 qOut

Figure 3.12: Translog cost function: Marginal costs depending on output quantity The resulting graphs are shown in figure 3.12. There is no clear relationship between marginal costs and the output quantity. Now, we illustrate our estimated model by drawing the average cost curve and the marginal cost curve for output quantities between 0 and five times the maximum output level in the sample, where we use the sample means of the input prices. > y lpCap lpLab lpMat totalCost margCost # average costs > plot( y, totalCost/y, type = "l" )

206

3 Dual Approach: Cost Functions > # marginal costs > lines( y, margCost, lty = 2 ) > # minimum and maximum output level in the sample > abline( v = min( dat$qOut ) ) > abline( v = max( dat$qOut ) ) > legend( "topright", lty = c( 1, 2 ), +

legend = c( "average costs", "marginal costs" ) )

> # average costs > plot( y, totalCost/y, type = "l", ylim = c( 0.07, 0.10 ) ) > # marginal costs > lines( y, margCost, lty = 2 ) > # maximum output level in the sample > abline( v = max( dat$qOut ) ) > legend( "topright", lty = c( 1, 2 ),

0.0e+00

4.0e+07

8.0e+07

1.2e+08

y

0.100 0.080

0.090

average costs marginal costs

0.070

0.1

0.2

0.3

0.4

average costs marginal costs

average costs, marginal costs

0.5

legend = c( "average costs", "marginal costs" ) )

average costs, marginal costs

+

0.0e+00

4.0e+07

8.0e+07

1.2e+08

y

Figure 3.13: Translog cost function: marginal and average costs The resulting graphs are shown in figure 3.13. The average costs are decreasing until an output level of around 70,000,000 units (1 unit ≈ 1 Euro) and they are increasing for larger output quantities. The average cost curve intersects the marginal cost curve (of course) at its minimum. However, as the maximum output level in the sample (approx. 25,000,000 units) is considerably lower than the minimum of the average cost curve (approx. 70,000,000 units), the estimated minimum of the average cost curve cannot be reliably determined because there are no data in this region.

207

3 Dual Approach: Cost Functions

3.5.8 Derived input demand functions We can derive the cost-minimizing input quantities from the Translog cost function using Shepard’s lemma: ∂c(w, y) ∂wi ∂ ln c(w, y) c = ∂ ln wi wi

xi (w, y) =



= αi +

N X

(3.108) (3.109) 

αij ln wj + αiy ln y 

j=1

c wi

(3.110)

And we can re-arrange these derived input demand functions in order to obtain the cost-minimizing cost shares: si (w, y) ≡

N X wi xi (w, y) = αi + αij ln wj + αiy ln y c j=1

(3.111)

We can calculate the cost-minimizing cost shares based on our estimated Translog cost function by following commands: > dat$shCap dat$shLab dat$shMat hist( dat$shCap ) > hist( dat$shLab ) > hist( dat$shMat ) The resulting graphs are shown in figure 3.14. As the signs of the derived optimal cost shares are equal to the signs of the first derivatives of the cost function with respect to the input prices, we can check whether the cost function is non-decreasing in input prices by checking if the derived optimal cost shares are non-negative. Counting the negative derived optimal cost shares, we find that our estimated cost function is decreasing in the capital price at 24 observations, decreasing in the labor price at 10 observations, and decreasing in the materials price at 3 observations. Given that out data set has 140 observations, our estimated cost function is to a large extent non-decreasing in input prices. As our estimated cost function is (forced to be) linearly homogeneous in all input prices, the derived optimal cost shares always sum up to one: > range( with( dat, shCap + shLab + shMat ) )

208

25 20 15 10

Frequency

−0.1 0.0

0.1

0.2

0.3

0.4

0

0

0

5

5

10

20

Frequency

15 10

Frequency

20

30

30

25

3 Dual Approach: Cost Functions

0.0

0.5

shCap

1.0

−0.2

0.2

shLab

0.4

0.6

0.8

1.0

shMat

Figure 3.14: Translog cost function: cost-minimizing cost shares [1] 1 1 We can use the following commands to compare the observed cost shares with the derived cost-minimizing cost shares: > compPlot( dat$shCap, dat$vCap / dat$cost ) > compPlot( dat$shLab, dat$vLab / dat$cost ) > compPlot( dat$shMat, dat$vMat / dat$cost )

shMat

0.6 0.2

observed

0.5 0.0

observed

1.0

0.1 0.2 0.3 0.4

1.0

shLab

−0.2

−0.1

observed

shCap

−0.1 0.0

0.1

0.2

0.3

0.4

0.0

optimal

0.5

1.0

optimal

−0.2 0.0

0.2

0.4

0.6

0.8

1.0

optimal

Figure 3.15: Translog cost function: observed and cost-minimizing cost shares The resulting graphs are shown in figure 3.15. Most firms use less than optimal materials, while there is a tendency to use more than optimal capital and a very slight tendency to use more than optimal labor. Similarly, we can compare the observed input quantities with the cost-minimizing input quantities: > compPlot( dat$shCap * dat$costTLHom / dat$pCap, +

dat$vCap / dat$pCap )

209

3 Dual Approach: Cost Functions > compPlot( dat$shLab * dat$costTLHom / dat$pLab, +

dat$vLab / dat$pLab )

> compPlot( dat$shMat * dat$costTLHom / dat$pMat, dat$vMat / dat$pMat )

qMat

observed

5e+05

−1e+05

1e+05

3e+05

5e+05

0

0e+00

observed

2e+05 4e+05

1e+06

200000

qLab

−1e+05

observed

qCap

100000

+

0e+00

5e+05

optimal

1e+06

0

optimal

50000

150000

250000

optimal

Figure 3.16: Translog cost function: observed and cost-minimizing input quantities The resulting graphs are shown in figure 3.16. Of course, the conclusions derived from these graphs are the same as conclusions derived from figure 3.15.

3.5.9 Derived input demand elasticities Based on the derived input demand functions (3.110), we can derive the input demand elasticities with respect to input prices: ij (w, y) =

∂xi (w, y) wj ∂wj xi (w, y) ∂

=



αi +

(3.112)

PN

k=1 αik

ln wk + αiy ln y



c wi



wj xi

∂wj "

=

N X αij c + αi + αik ln wk + αiy ln y wj wi k=1

−δij αi +

N X

!

αik ln wk + αiy ln y

k=1

"

!

xj wi

(3.113) (3.114)

#

c wj wi2 xi

#

αij c x i wi x j x i wi c wj = + − δij wi wj c wi c wi2 xi αij c wj x j wj + − δij = wi x i c wi αij = + sj − δij , si

210

(3.115) (3.116) (3.117)

3 Dual Approach: Cost Functions where δij (again) denotes Kronecker’s delta (2.95), and the input demand elasticities with respect to the output quantity: iy (w, y) =

y ∂xi (w, y) ∂y xi (w, y) ∂



=

αi +

(3.118)

PN

k=1 αik

ln wk + αiy ln y



c wi



∂y "

=

N X αiy c + αi + αik ln wk + αiy ln y y wi k=1

αiy c wi xi ∂c 1 y = + wi y c ∂y wi xi αiy c ∂c y = + wi xi ∂y c αiy ∂ ln c = + , si ∂ ln y 

y xi !

(3.119) #

∂c 1 y ∂y wi xi

(3.120)



(3.121) (3.122) (3.123)

where ∂ ln c/∂ ln y is the cost flexibility (see section 3.1.3). With the following commands, we compute the input demand elasticities at the first observation: > ela rownames( ela ) colnames( ela ) ela[ 1, 1 ] ela[ 1, 2 ] ela[ 1, 3 ] ela[ 1, 4 ] ela[ 2, 1 ] ela[ 2, 2 ] ela[ 2, 3 ] ela[ 2, 4 ] ela[ 3, 1 ] ela[ 3, 2 ] ela[ 3, 3 ] ela[ 3, 4 ] ela wCap qCap -0.6383107

wLab

wMat

y

-1.1835104 1.821821136 0.1653394

qLab

4.4484591 -11.3084023 6.859943173 0.2293691

qMat

0.5938258

-0.5948896 0.001063746 0.3983336

These demand elasticities indicate that when the capital price increases by one percent, the

211

3 Dual Approach: Cost Functions demand for capital decreases by 0.638 percent, the demand for labor increases by 4.448 percent, and the demand for materials increases by 0.594 percent. When the labor price increases by one percent, the elasticities indicate that the demand for all inputs decreases, which is not possible when the output quantity should be maintained. Furthermore, the symmetry condition for the elasticities (3.69) indicates that the cross-price elasticities of each input pair must have the same sign. However, this is not the case for the pairs capital–labor and materials–labor. The reason for this is the negative predicted input share of labor: > dat[ 1, c( "shCap", "shLab", "shMat" ) ] shCap

shLab

shMat

1 0.2630271 -0.06997825 0.8069511 Finally, the negativity constraint (3.67) is violated, because the own-price elasticity of materials is positive (0.001). When the output quantity is increased by one percent, the demand for capital increases by 0.165 percent, the demand for labor increases by 0.229 percent, and the demand for materials increases by 0.398 percent. Now, we create a three-dimensional array and compute the demand elasticities for all observations: > elaAll dimnames( elaAll ) elaAll[ 1, 1, ] elaAll[ 1, 2, ] elaAll[ 1, 3, ] elaAll[ 1, 4, ] elaAll[ 2, 1, ] elaAll[ 2, 2, ] elaAll[ 2, 3, ] elaAll[ 2, 4, ] elaAll[ 3, 1, ] elaAll[ 3, 2, ] elaAll[ 3, 3, ] elaAll[ 3, 4, ] dat$monoCostTLHom = 0 & shLab >= 0 & shMat >= 0 & +

costFlex >= 0 )

> apply( elaAll[ , , dat$monoCostTLHom ], c(1,2), median )

212

3 Dual Approach: Cost Functions wCap

wLab

qCap -0.6486401 -2.0250830 qLab -0.7529717 qMat

wMat

y

2.6857495 -0.007520786

1.4426349 -0.6992232

0.467598439

0.6641220 -0.5086388 -0.1645389

0.506421497

We can visualize the variations of the elasticities using histograms, where we—again—only include observations, at which the monotonicity conditions are fulfilled. > hist( elaAll[1,1,dat$monoCostTLHom] ) > hist( elaAll[1,2,dat$monoCostTLHom] ) > hist( elaAll[1,3,dat$monoCostTLHom] ) > hist( elaAll[2,1,dat$monoCostTLHom] ) > hist( elaAll[2,2,dat$monoCostTLHom] ) > hist( elaAll[2,3,dat$monoCostTLHom] ) > hist( elaAll[3,1,dat$monoCostTLHom] ) > hist( elaAll[3,2,dat$monoCostTLHom] ) > hist( elaAll[3,3,dat$monoCostTLHom] ) > hist( elaAll[1,4,dat$monoCostTLHom] ) > hist( elaAll[2,4,dat$monoCostTLHom] ) > hist( elaAll[3,4,dat$monoCostTLHom] ) The resulting graphs are shown in figure 3.17. While the conditional own-price elasticities of capital and materials are negative at almost all observations, the conditional own-price elasticity of labor is positive at almost all observations. These violations of the negativity constraint (3.67) originate from the violation of the concavity condition. As all conditional elasticities of the capital demand with respect to the materials price as well as all conditional elasticities of the materials demand with respect to the capital price are positive, we can conclude that capital and materials are net substitutes. In contrast, all cross-price elasticities between capital and labor as well as between labor and materials are negative. This indicates that the two pairs capital and labor as well as labor and materials are net complements. When the output quantity is increased by one percent, most farms would increase both the labor quantity and the materials quantity by around 0.5% and either increase or decrease the capital quantity.

3.5.10 Theoretical consistency The Translog cost function (3.86) is always continuous for positive input prices and a positive output quantity. The non-negativity is always fulfilled for the Translog cost function, because the predicted cost is equal to the exponential function of the right-hand side of equation (3.86) and the exponential function always returns a non-negative value (also when the right-hand side of equation (3.86) is negative).

213

10

15

20

100 −200

0

0

−20

−10

0

100

150

200

80 Frequency

20 0 0

20

40

60

80

100

−70

−50

E lab lab

−30

−10 0

E lab mat

1

2

3

4

5

6

0

20

40

Frequency

60

80

80 60 0

0

20

40

Frequency

60 40 20

Frequency

80

100

E lab cap

0

7

−8

−6

−4

−2

0

−0.5

0.5

1.5

2.5

E mat mat

−20

−10

0

30 10 0

0 −30

20

Frequency

10

20

Frequency

30 20 10 0

Frequency

30

40

40

40

50

E mat lab

50

E mat cap

−40

250

100

100 80 60 20 0 −30

50

E cap mat

40

Frequency

80 60 40 0

20

Frequency

−100

E cap lab

100

E cap cap

−40

60 20 0

−300

60

5

40

0

40

Frequency

80

80 60 0

0

20

40

Frequency

60 40 20

Frequency

80

100

3 Dual Approach: Cost Functions

0.0

0.5

E cap y

1.0

1.5

0.0

0.4

E lab y

Figure 3.17: Translog cost function: derived demand elasticities

214

0.8 E mat y

1.2

3 Dual Approach: Cost Functions If the output quantity approaches zero (from above), the right-hand side of the Translog cost functions (equation 3.86) approaches: 



N X

N X N N X 1X 1 lim α0 + αi ln wi + αy ln y + αiy ln wi ln y  αij ln wi ln wj + αyy (ln y)2 + y→0+ 2 2 i=1 i=1 j=1 i=1

(3.124) = lim

y→0+

N X 1 αy ln y + αyy (ln y)2 + αiy ln wi ln y 2 i=1 N X 1 αiy ln wi ln y αy + αyy ln y + 2 i=1

!

= lim

y→0+

= lim

y→0+

N X 1 αy + αyy ln y + αiy ln wi 2 i=1

!

!

(3.126)

!

lim ln y

y→0+

= lim (αyy ln y) lim ln y = (−αyy ∞)(−∞) = αyy ∞ y→0+

(3.125)

y→0+

(3.127) (3.128)

Hence, if coefficient αyy is negative and the output quantity approaches zero (from above), the predicted cost (exponential function of the right-hand side of equation 3.86) approaches zero so that the “no fixed costs” property is asymptotically fulfilled. Our estimated Translog cost function with linear homogeneity in input prices imposed (of course) is linearly homogeneous in input prices. Hence, the linear homogeneity property is globally fulfilled. A cost function is non-decreasing in the output quantity if the cost flexibility and the elasticity of size are non-negative. As we can see from figure 3.8, only a single cost flexibility and thus, only a single elasticity of size is negative. Hence, our estimated Translog cost function with linear homogeneity in input prices imposed violates the monotonicity condition regarding the output quantity only at a single observation. Given Shepard’s lemma, a cost function is non-decreasing in input prices if the derived costminimizing input quantities and the corresponding cost shares are non-negative. As we can see from figure 3.14, our estimated Translog cost function with linear homogeneity in input prices imposed predicts that 24 cost shares of capital, 10 cost shares of labor, and 3 cost shares of materials are negative. In total, the monotonicity condition regarding the input prices is violated at 36 observations: > table( dat$shCap >= 0 & dat$shLab >= 0 & dat$shMat >= 0 ) FALSE

TRUE

36

104

Concavity in input prices of the cost function requires that the Hessian matrix of the cost function with respect to the input prices is negative semidefinite. The elements of the Hessian

215

3 Dual Approach: Cost Functions matrix are: Hij =

∂xi (w, y) ∂ 2 c(w, y) = ∂wi ∂wj ∂wj c wi



N X αij c = + αi + αik ln wk + αiy ln y wj wi k=1

!

∂ =



(3.129)

αi +

PN

k=1 αik

ln wk + αiy ln y



(3.130)

∂wj N X xj − δij αi + αik ln wk + αiy ln y wi k=1

!

c wi2 (3.131)

x i wi x j x i wi c αij c + − δij wi wj c wi c wi2 xi xj xi αij c + − δij , = wi wj c wi

(3.132)

=

(3.133)

where δij (again) denotes Kronecker’s delta (2.95). As the elements of the Hessian matrix have the same sign as the corresponding elasticities (Hij = ij (w, y) xi /wj ), the positive own-price elasticities of labor in figure 3.17 indicate that the element Hlab,lab is positive at all observations, where the monotonicity conditions regarding the input prices are fulfilled. As negative semidefiniteness requires that all diagonal elements of the (Hessian) matrix are negative, we can conclude that the estimated Translog cost function is concave at not a single observation where the monotonicity conditions regarding the input prices are fulfilled. This means that our estimated Translog cost function is inconsistent with microeconomic theory at all observations.

3.6 Translog cost function with multiple outputs 3.6.1 Specification The Translog cost function with multiple outputs is a minor extension of the Translog cost function with a single output as defined in equation (3.86): ln c(w, y) = α0 + +

+

1 2

N X

αi ln wi +

i=1 N X N X

M X

βm ln ym

m=1

αij ln wi ln wj +

i=1 j=1

N X M X

γim ln wi ln ym

i=1 m=1

with αij = αji ∀ i, j and βml = βlm ∀ m, l.

216

M X M 1 X βml ln ym ln yl 2 m=1 l=1

(3.134)

3 Dual Approach: Cost Functions

3.6.2 Properties The Translog multiple-output cost function cannot be globally quasi-convex in output quantities because it cannot have globally concave iso-cost curves for different output-output combinations. In fact, both the Cobb-Douglas and the Translog multiple-output cost functions predict that when decreasing one output quantity towards zero, the other output quantities can be increased towards infinity while still holding total costs costant.

217

4 Dual Approach: Profit Function 4.1 Theory 4.1.1 Profit functions The profit function: π(p, w) = max p y − y,x

X

wi xi , s.t. y = f (x)

(4.1)

i

returns the maximum profit that is attainable given the output price p and input prices w. It is important to distinguish the profit definition (1.4) from the profit function (4.1).

4.1.2 Short-run profit functions As producers often cannot instantly adjust the quantity of the some inputs (e.g. capital, land, apple trees), estimating a short-run profit function with some quasi-fixed input quantities might be more appropriate than a (long-run) profit function which assumes that all input quantities and output quantities can be adjusted instantly. Furthermore, a short-run profit function can model technologies with increasing returns to scale, if the sum over the output elasticities of the variable inputs is lower than one. In general, a short-run profit function is defined as n

o

π v (p, w1 , x2 ) = max p y − cs (w1 , y, x2 ) , y≥0

(4.2)

where w1 denotes the vector of the prices of all variable inputs, x2 denotes the vector of the quantities of all quasi-fixed inputs, cs (w1 , y, x2 ) is the short-run cost function (see section 3.3), π v denotes the gross margin defined in equation (1.5), and N 1 is a vector of the indices of the variable inputs.

4.2 Graphical illustration of profit and gross margin We use the following commands to visualize the variation of the profits and the relationship between profits and firm size: > hist( dat$profit, 30 ) > plot( dat$X, dat$profit, log = "xy" )

218

1e+07

0

2e+04

5e+05

profit

60 40 20

Frequency

80

4 Dual Approach: Profit Function

0e+00

2e+07

4e+07

6e+07

0.5

profit

1.0

2.0

5.0

X

Figure 4.1: Profit The resulting graphs are shown in figure 4.1. The histogram shows that 14 out of 140 apple producers (10%) have (slightly) negative profits. Although this seems to be not unrealistic, this contradicts the non-negativity condition of the profit function. However, the observed negative profits might have been caused by deviations from the theoretical assumptions that we have made to derive the profit function, e.g. that all inputs can be instantly adjusted and that there are no unexpected events such as severe weather conditions or pests. We will deal with these deviations from our assumptions later and for now just ignore the observations with negative profits in our analyses with the profit function. The right part of figure 4.1 shows that the profit clearly increases with firm size. The following commands graphically illustrate the variation of the gross margins and their relationship to the firm size and the quantity of the quasi-fixed input: > hist( dat$vProfit, 30 ) > plot( dat$X, dat$vProfit, log = "xy" ) > plot( dat$qCap, dat$vProfit, log = "xy" ) The resulting graphs are shown in figure 4.2. The histogram on the left shows that 8 out of 140 apple producers (6%) have (slightly) negative gross margins. Although this does not seem to be unrealistic, this contradicts the non-negativity condition of the short-run profit function. However, the observed negative gross margins might have been caused by deviations from the theoretical assumptions, e.g. that there are no unexpected events such as severe weather conditions or pests. The center part of figure 4.2 shows that the gross margin clearly increases with the firm size (as expected). However, the right part of this figure shows that the gross margin is only weakly positively correlated with the fixed input. Please note that according to microeconomic theory, the short-run total profit π s —in contrast

219

0e+00

2e+07

4e+07

6e+07

0.5

1.0

gross margin

2.0

5.0

X

5e+03 5e+04 5e+05 5e+06 5e+07

gross margin

5e+03 5e+04 5e+05 5e+06 5e+07

gross margin

60 40 0

20

Frequency

80

4 Dual Approach: Profit Function

5e+03

2e+04

1e+05

5e+05

qCap

Figure 4.2: Gross margins to the gross margin π v —might be negative due to fixed costs: π s (p, w, x2 ) = π v (p, w1 , x2 ) −

X

wj x j ,

(4.3)

j∈N 2

where N 2 is a vector of the indices of the quasi-fixed inputs. However, in the long-run, profit must be non-negative: π(p, w) = max π s (p, w, x2 ) ≥ 0, x2

(4.4)

as all costs are variable in the long run.

4.3 Cobb-Douglas profit function 4.3.1 Specification The Cobb-Douglas profit function1 has the following specification: ! αp

π = Ap

Y α w i i

,

(4.5)

i

This function can be linearized to ln π = α0 + αp ln p +

X

αi ln wi

(4.6)

i

with α0 = ln A.

1

Please note that the Cobb-Douglas profit function is used as a simple example here but that it is much too restrictive for most “real” empirical applications (Chand and Kaul, 1986).

220

4 Dual Approach: Profit Function

4.3.2 Estimation The linearized Cobb-Douglas profit function can be estimated by OLS. As the logarithm of a negative number is not defined and function lm automatically removes observations with missing data, we do not have to remove the observations (apple producers) with negative profits manually. > profitCD summary( profitCD ) Call: lm(formula = log(profit) ~ log(pOut) + log(pCap) + log(pLab) + log(pMat), data = dat) Residuals: Min

1Q

Median

3Q

Max

-3.6183 -0.2778

0.1261

0.5986

2.0442

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

13.9380

0.4921

28.321

< 2e-16 ***

log(pOut)

2.7117

0.2340

11.590

< 2e-16 ***

log(pCap)

-0.7298

0.1752

-4.165 5.86e-05 ***

log(pLab)

-0.1940

0.4623

-0.420

0.676

log(pMat)

0.1612

0.2543

0.634

0.527

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9815 on 121 degrees of freedom (14 observations deleted due to missingness) Multiple R-squared:

0.5911,

F-statistic: 43.73 on 4 and 121 DF,

Adjusted R-squared:

0.5776

p-value: < 2.2e-16

As expected, lm reports that 14 observations have been removed due to missing data (logarithms of negative numbers).

4.3.3 Properties A Cobb-Douglas profit function is always continuous and twice continuously differentiable for all p > 0 and wi > 0 ∀i. Furthermore, a Cobb-Douglas profit function automatically fulfills the non-negativity property, because the profit predicted by equation (4.5) is always positive as long as coefficient A is positive (given that all input prices and the output price are positive). As A

221

4 Dual Approach: Profit Function is usually obtained by applying the exponential function to the estimate of α0 , i.e. A = exp(α0 ), A and hence, also the predicted profit, are always positive (even if α0 is non-positive). The estimated coefficients of the output price and the input prices indicate that profit is increasing in the output price and decreasing in the capital and labor price but it is increasing in the price of materials, which contradicts microeconomic theory. However, the positive coefficient of the (logarithmic) price of materials is statistically not significantly different from zero. The Cobb-Douglas profit function is linearly homogeneous in all prices (output price and all input prices) if the following condition is fulfilled: t π(p, w) = π(t p, t w)

(4.7)

ln(t π) = α0 + αp ln(t p) +

X

αi ln(t wi )

(4.8)

i

ln t + ln π = α0 + αp ln t + αp ln p +

X

αi ln t +

i

X

αi ln wi

(4.9)

i

!

ln t + ln π = α0 + αp ln p +

X

αi ln wi + ln t αp +

i

X

αi

(4.10)

i

!

ln π = ln π + ln t

αp +

X

αi − 1

(4.11)

i

!

0 = ln t

αp +

X

αi − 1

(4.12)

i

0 = αp +

X

αi − 1

(4.13)

αi

(4.14)

i

1 = αp +

X i

Hence, the homogeneity condition is only fulfilled if the coefficient of the (logarithmic) output price and the coefficients of the (logarithmic) input prices sum up to one. As they sum up to 2.71 + (−0.73) + (−0.19) + 0.16 = 1.95, the homogeneity condition is not fulfilled in our estimated model.

4.3.4 Estimation with linear homogeneity in all prices imposed In order to derive a Cobb-Douglas profit function with linear homogeneity in input prices imposed, we re-arrange the homogeneity condition (4.14) to get αp = 1 −

N X

αi

(4.15)

i=1

and replace αp in the profit function (4.6) by the right-hand side of the above equation: !

ln π = α0 + 1 −

X i

222

αi ln p +

X i

αi ln wi

(4.16)

4 Dual Approach: Profit Function ln π = α0 + ln p −

X

αi ln p +

X

i

ln π − ln p = α0 +

X

αi ln wi

(4.17)

i

αi (ln wi − ln p)

(4.18)

i

ln

X π wi = α0 + αi ln p p i

(4.19)

This Cobb-Douglas profit function with linear homogeneity imposed can be estimated by following command: > profitCDHom summary( profitCDHom ) Call: lm(formula = log(profit/pOut) ~ log(pCap/pOut) + log(pLab/pOut) + log(pMat/pOut), data = dat) Residuals: Min

1Q

Median

3Q

Max

-3.6045 -0.2724

0.0972

0.6013

2.0385

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

14.27961

0.45962

31.068

< 2e-16 ***

log(pCap/pOut) -0.82114

0.16953

-4.844 3.78e-06 ***

log(pLab/pOut) -0.90068

0.25591

-3.519 0.000609 ***

log(pMat/pOut) -0.02469

0.23530

-0.105 0.916610

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9909 on 122 degrees of freedom (14 observations deleted due to missingness) Multiple R-squared:

0.3568,

F-statistic: 22.56 on 3 and 122 DF,

Adjusted R-squared:

0.341

p-value: 1.091e-11

The coefficient of the (logarithmic) output price can be obtained by the homogeneity restriction (4.15). Hence, it is 1 − (−0.82) − (−0.9) − (−0.02) = 2.75. Now, all monotonicity conditions are fulfilled: profit is increasing in the output price and decreasing in all input prices. We can use a Wald test or a likelihood-ratio test to test whether the model and the data contradict the homogeneity assumption:

223

4 Dual Approach: Profit Function > library( "car" ) > linearHypothesis( profitCD, "log(pOut) + log(pCap) + log(pLab) + log(pMat) = 1" ) Linear hypothesis test Hypothesis: log(pOut)

+ log(pCap)

+ log(pLab)

+ log(pMat) = 1

Model 1: restricted model Model 2: log(profit) ~ log(pOut) + log(pCap) + log(pLab) + log(pMat) Res.Df

RSS Df Sum of Sq

1

122 119.78

2

121 116.57

1

F

Pr(>F)

3.2183 3.3407 0.07005 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> lrtest( profitCD, profitCDHom ) Likelihood ratio test Model 1: log(profit) ~ log(pOut) + log(pCap) + log(pLab) + log(pMat) Model 2: log(profit/pOut) ~ log(pCap/pOut) + log(pLab/pOut) + log(pMat/pOut) #Df

LogLik Df

Chisq Pr(>Chisq)

1

6 -173.88

2

5 -175.60 -1 3.4316

0.06396 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Both tests reject the null hypothesis, linear homogeneity in all prices, at the 10% significance level but not at the 5% level. Given the importance of microeconomic consistency and that 5% is the standard significance level, we continue our analysis with the Cobb-Douglas profit function with linear homogeneity imposed.

4.3.5 Checking convexity in all prices The last property that we have to check is the convexity in all prices. A continuous and twice continuously differentiable function is convex, if its Hessian matrix is positive semidefinite. A necessary and sufficient condition for positive semidefiniteness is that all its principal minors (not only its leading principal minors) are non-negative (see section 1.5.2). The first derivatives of the

224

4 Dual Approach: Profit Function Cobb-Douglas profit function with respect to the input prices are: π ∂π ∂ ln π π = = αi ∂wi ∂ ln wi wi wi

(4.20)

and the first derivative with respect to the output price is: π ∂ ln π π ∂π = = αp ∂p ∂ ln p p p

(4.21)

Now, we can calculate the second derivatives as derivatives of the first derivatives (4.20) and (4.21): ∂2π ∂wi ∂wj

∂2π ∂wi ∂p

=

∂π ∂ ∂w i



∂ αi wπi



= ∂wj ∂wj αi ∂π π = − δij αi 2 wi ∂wj wi π π αi = αj − δij αi 2 wi wj wi π = αi (αj − δij ) wi wj =

∂π ∂ ∂w i



∂ αi wπi

= ∂p αi ∂π = wi ∂p αi π = αp wi p π = αi αp wi p

∂p

(4.22) (4.23) (4.24) (4.25)



(4.26) (4.27) (4.28) (4.29)





∂ αp πp ∂ ∂π ∂2π ∂p = = ∂p2 ∂p ∂p π αp ∂π = − αp 2 p ∂p p αp π π = αp − αp 2 p p p π = αp (αp − 1) 2 , p

(4.30) (4.31) (4.32) (4.33)

where δij (again) denotes Kronecker’s delta (2.95). As all elements of the Hessian matrix include π as a multiplicative term, we can ignore this variable in the calculation of the Hessian matrix, because the value π neither changes the signs of the (diagonal) elements of the matrix, nor the signs of the principal minors and the determinant (as long as π is positive, i.e. the non-negativity condition is fulfilled) given the general rule that |π · M | = π · |M |, where M denotes a quadratic matrix, π denotes a scalar, and the two vertical bars denote the determinant function.

225

4 Dual Approach: Profit Function We start with checking convexity in all prices of the Cobb-Douglas profit function without homogeneity imposed. To simplify the calculations, we define short-cuts for the coefficients: > gCap gLab gMat gOut hpCapCap hpLabLab hpMatMat hpCapLab hpCapMat hpLabMat hpCapOut hpLabOut hpMatOut hpOutOut hessian hessian[ 1, 1 ] hessian[ 2, 2 ] hessian[ 3, 3 ] hessian[ 1, 2 ] det( hessianHom[-2,-2] ) [1] 0.0002138007 > det( hessianHom[-1,-1] ) [1] 0.001795727 > det( hessianHom ) [1] 3.268657e-19 These principal minors are all either positive or very close to zero. For instance, the determinant (fourth-order principal minor) of the Hessian is close to zero, where it is negative on some computers but positive on other computers. As Hessian matrices of linear homogeneous functions are always singular, it is expected that the determinant of the Hessian matrix (the N th-order principal minor) is zero. However, the computed determinant of our Hessian matrix is not exactly zero due to rounding errors, which are unavoidable on digital computers. Rounding errors may also have caused that our calculations of other principle minors resulted in values that are negative but very close to zero, although they are in fact zero or slightly positive. Assuming that the principal minors that are very close to zero are in fact zero or positive, we can conclude that our Cobb-Douglas profit function with linear homogeneity imposed is convex in all prices at the first observation. We can check the semidefiniteness of a matrix more conveniently with the command semidefiniteness (package miscTools), which (by default) checks the signs of the principal minors and returns a logical value indicating whether the necessary and sufficient conditions for negative or positive semidefiniteness are fulfilled: We can check the semidefiniteness of a matrix more conveniently with the command semidefiniteness (package miscTools), which (by default) checks the signs of the principal minors and returns a logical value indicating whether the necessary and sufficient conditions for negative or positive semidefiniteness are fulfilled: > semidefiniteness( hessianHom, positive = TRUE ) [1] TRUE In the following, we will check whether convexity in all prices is fulfilled at each observation in the sample:

229

4 Dual Approach: Profit Function > dat$convexCDHom for( obs in 1:nrow( dat ) ) { +

hessianLoop profitCDSRHom summary( profitCDSRHom ) Call: lm(formula = log(vProfit/pOut) ~ log(pLab/pOut) + log(pMat/pOut) + log(qCap), data = dat) Residuals: Min

1Q

Median

3Q

Max

-4.7302 -0.0677

0.2598

0.5160

0.8916

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

3.3145

1.2184

log(pLab/pOut)

-1.4574

0.2252

-6.471 1.88e-09 ***

log(pMat/pOut)

-0.7156

0.2427

-2.949

1.0847

0.1212

log(qCap)

2.720

0.00743 ** 0.00380 **

8.949 3.50e-15 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.9628 on 128 degrees of freedom (8 observations deleted due to missingness) Multiple R-squared:

0.5227,

F-statistic: 46.73 on 3 and 128 DF,

Adjusted R-squared:

0.5115

p-value: < 2.2e-16

We can obtain the coefficient of the output price from the homogeneity condition (4.15): 1 − (−1.457) − (−0.716) = 3.173. All microeconomic monotonicity conditions are still fulfilled: the Cobb-Douglas short-run profit function with homogeneity imposed is increasing in the output price, decreasing in the prices of all variable inputs, and increasing in the quasi-fixed input quantity. We can test the homogeneity restriction by a likelihood ratio test: > lrtest( profitCDSRHom, profitCDSR ) Likelihood ratio test Model 1: log(vProfit/pOut) ~ log(pLab/pOut) + log(pMat/pOut) + log(qCap) Model 2: log(vProfit) ~ log(pOut) + log(pLab) + log(pMat) + log(qCap) #Df

LogLik Df

1

5 -180.27

2

6 -180.17

Chisq Pr(>Chisq)

1 0.1859

0.6664

237

4 Dual Approach: Profit Function Given the large P -value, we can conclude that the data do not contradict the linear homogeneity in the output price and the prices of the variable inputs.

4.4.5 Returns to scale The sum over the coefficients of all quasi-fixed inputs indicates the percentage change of the gross margin if the quantities of all quasi-fixed inputs are increased by one percent: If this sum is larger than one, the increase in gross margin is more than proportional to the increase in the quasi-fixed inputs. Hence, the technology has increasing returns to scale. If this sum over the coefficients of all quasi-fixed inputs is smaller than one, the increase in gross margin is less than proportional to the increase in the quasi-fixed inputs and the technology has decreasing returns to scale. As the coefficient of our (single) quasi-fixed input is larger than one (1.085), we can conclude that the technology has increasing returns to scale.

4.4.6 Shadow prices of quasi-fixed inputs The partial derivatives of the short-run profit function with respect to the quantities of the quasi-fixed inputs denote the additional gross margins that can be earned by an additional unit of these quasi-fixed inputs. These internal marginal values of the quasi-fixed inputs are usually called “shadow prices”. In case of the Cobb-Douglas short-run profit function, the shadow prices can be computed by

∂π v ∂ ln π v π v πv = = αj ∂xj ∂ ln xj xj xj

(4.68)

Before we can calculate the shadow price of the capital input, we need to calculate the predicted gross margin π v . As the dependent variable of the Cobb-Douglas short-run profit function with homogeneity imposed is ln(π v / ln p), we have to apply the exponential function to the fitted dependent variable and then we have to multiply the result with p, in order obtain the fitted gross margins π v . Furthermore, we have to be aware of that the fitted method only returns the predicted values for the observations that were included in the estimation. Hence, we have to make sure that the predicted gross margins are only assigned to the observations that have a positive gross margin and hence, were included in the estimation: > dat$vProfitCDHom[ dat$vProfit > 0 ] 0 ]

Now, we can calculate the shadow price of the capital input for each apple producer who has a positive gross margin and hence, was included in the estimation: > dat$pCapShadow hist( dat$pCapShadow, 30 ) > hist( dat$pCapShadow[ dat$pCapShadow < 30 ], 30 )

0

100

200

300

400

shadow price of capital

50.0 5.0 0.2

1.0

shadow prices

6 0

0

2

4

Frequency

40 30 20 10

Frequency

8

50

10

60

500.0

> compPlot( dat$pCap, dat$pCapShadow, log = "xy" )

0

5

10

15

20

25

30

shadow price of capital

0.2

1.0

5.0 20.0

200.0

observed prices

Figure 4.4: Shadow prices of capital The resulting graphs are shown in figure 4.4. The two histograms show that most shadow prices are below 30 and many shadow prices are between 3 and 11 but there are also some apple producers who would gain much more from increasing their capital input. Indeed, all apple producers have a higher shadow price of capital than the observed price of capital, where the difference is small for some producers and large for other producers. These differences can be explained by risk aversion and market failures on the credit market or land market (e.g. marginal prices are not equal to average prices).

239

5 Efficiency Measures 5.1 Technical efficiency 5.1.1 Output-oriented technical efficiency with one output The output-oriented technical efficiency according to Shepard is defined as TE =

y y∗

⇔

y = T E · y∗

0 ≤ T E ≤ 1,

(5.1)

where y is the observed output quantity and y ∗ is the maximum output quantity that can be produced with the observed input quantities x. The output-oriented technical efficiency according to Farrell is defined as y∗ y

TE =

⇔

y∗ = T E · y

T E ≥ 1.

(5.2)

These efficiency measures are graphically illustrated in Bogetoft and Otto (2011, p. 26, figure 2.2).

5.1.2 Input-oriented technical efficiency with one input The input-oriented technical efficiency according to Shepard is defined as TE =

x x∗

⇔

x = T E · x∗

T E ≥ 1,

(5.3)

where x is the observed input quantity and x∗ is the minimum input quantity at which the observed output quantities y can be produced. The input-oriented technical efficiency according to Farrell is defined as TE =

x∗ x

⇔

x∗ = T E · x

0 ≤ T E ≤ 1.

(5.4)

These efficiency measures are graphically illustrated in Bogetoft and Otto (2011, p. 26, figure 2.2).

5.1.3 Output-oriented technical efficiency with two or more outputs The output-oriented technical efficiencies according to Shepard and Farrell assume a proportional increase of all output quantities, while all input quantities are held constant.

240

5 Efficiency Measures Hence, the output-oriented technical efficiency according to Shepard is defined as TE =

y1 y2 yM = ∗ = ... = ∗ ∗ y1 y2 yM

yi = T E · yi∗ ∀ i

⇔

0 ≤ T E ≤ 1,

(5.5)

∗ are the maximum output where y1 , y2 , . . . , yM are the observed output quantities, y1∗ , y2∗ , . . . , yM

quantities (given a proportional increase of all output quantities) that can be produced with the observed input quantities x, and M is the number of outputs. The output-oriented technical efficiency according to Farrell is defined as TE =

y1∗ y∗ y∗ = 2 = ... = M y1 y2 yM

⇔

yi∗ = T E · yi ∀ i

T E ≥ 1.

(5.6)

These efficiency measures are graphically illustrated in Bogetoft and Otto (2011, p. 27, figure 2.3, right panel).

5.1.4 Input-oriented technical efficiency with two or more inputs The input-oriented technical efficiencies according to Shepard and Farrell assume a proportional reduction of all inputs, while all outputs are held constant. Hence, the input-oriented technical efficiency according to Shepard is defined as TE =

x1 x2 xN = ∗ = ... = ∗ ∗ x1 x2 xN

⇔

xi = T E · x∗i ∀ i

TE ≥ 1

(5.7)

where x1 , x2 , . . . , xN are the observed input quantities, x∗1 , x∗2 , . . . , x∗N are the minimum input quantities (given a proportional decrease of all input quantities) at which the observed output quantities y can be produced, and N is the number of inputs. The input-oriented technical efficiency according to Farrell is defined as TE =

x∗ x∗ x∗1 = 2 = ... = N x1 x2 xN

⇔

x∗i = T E · xi ∀ i

0 ≤ T E ≤ 1.

(5.8)

These efficiency measures are graphically illustrated in Bogetoft and Otto (2011, p. 27, figure 2.3, left panel).

5.2 Allocative efficiency, revenue efficiency, cost efficiency 5.2.1 Output-oriented allocative efficiency and revenue efficiency According to equation (5.6), the output-oriented technical efficiency according to Farrell is TE =

y˜1 y˜2 y˜M p y˜ = = ... = = , y1 y2 yM py

(5.9)

where y˜ is the vector of technically efficient output quantities and p is the vector of output prices.

241

5 Efficiency Measures Proof of the last equality in equation (5.9): P

p y˜ pi y˜i = = Pi py i p i yi

P P p i yi i pi T E y i P = T E Pi = TE i pi

yi

i pi

yi

(5.10)

The output-oriented allocative efficiency according to Farrell is defined as AE =

p y∗ p yˆ = , p y˜ p y˜

(5.11)

where y ∗ is the vector of technically efficient and allocatively efficient output quantities and yˆ is the vector of output quantities so that p yˆ = p y ∗ and yˆi /˜ yi = AE ∀ i. Finally, the revenue efficiency according to Farrell is RE =

p y∗ p y ∗ p y˜ = = AE · T E py p y˜ p y

(5.12)

All these efficiency measures can also be specified according to Shepard by just taking the inverse of the Farrell specifications. These efficiency measures are graphically illustrated in Bogetoft and Otto (2011, p. 40, figure 2.11).

5.2.2 Input-oriented allocative efficiency and cost efficiency According to equation (5.8), the input-oriented technical efficiency according to Farrell is TE =

x ˜1 x ˜2 x ˜N wx ˜ = = ... = = , x1 x2 xN wx

(5.13)

where x ˜ is the vector of technically efficient input quantities and w is the vector of output prices. The input-oriented allocative efficiency according to Farrell is defined as AE =

w x∗ wx ˆ = , wx ˜ wx ˜

(5.14)

where x∗ is the vector of technically efficient and allocatively efficient input quantities and x ˆ is the vector of output quantities so that w x ˆ = w x∗ and x ˆi /˜ xi = AE ∀ i. Finally, the cost efficiency according to Farrell is CE =

w x∗ w x∗ w x ˜ = = AE · T E wx wx ˜ wx

(5.15)

All these efficiency measures can also be specified according to Shepard by just taking the inverse of the Farrell specifications. These efficiency measures are graphically illustrated in Bogetoft and Otto (2011, p. 36, figure 2.9).

242

5 Efficiency Measures

5.3 Profit efficiency The profit efficiency according to Farrell is defined as PE =

p y ∗ − w x∗ , py−w x

(5.16)

where y ∗ and x∗ denote the profit maximizing output quantities and input quantities, respectively (assuming full technical efficiency). The profit efficiency according to Shepard is just the inverse of the Farrell specifications.

5.4 Scale efficiency In case of one input x and one output y = f (x), the scale efficiency according to Farrell is defined as SE =

AP ∗ , AP

(5.17)

where AP = f (x)/x is the observed average product AP ∗ = f (x∗ )/x∗ is the maximum average product, and x∗ is the input quantity that results in the maximum average product. The first-order condition for a maximum of the average product is ∂f (x) 1 f (x) ∂AP = − 2 =0 ∂x ∂x x x

(5.18)

This condition can be re-arranged to ∂f (x) x =1 ∂x f (x)

(5.19)

Hence, a necessary (but not sufficient) condition for a maximum of the average product is an elasticity of scale equal to one.

243

6 Stochastic Frontier Analysis 6.1 Stochastic production frontiers 6.1.1 Specification In section 2, we have estimated average production functions, where about half of the observations were below the estimated production function and about half of the observations were above the estimated production function (see left panel of figure 6.1). However, in microeconomic theory, the production function indicates the maximum output quantity for each given set of input quantities. Hence, theoretically, no observation could be above the production function and an observations below the production function would indicate technical inefficiency. y

y

o

o o o

o o o

o

o

o

o

o

o

o

o

o

o o o

o

o

o o o

o

o o

o

o

o

o

o

o

o

x

x

Source: Bogetoft and Otto (2011) Figure 6.1: Production function estimation: ordinary regression and with intercept correction This means that all residuals must be negative or zero. A production function with only non-positive residuals could look like: ln y = ln f (x) − u with u ≥ 0,

(6.1)

where −u ≤ 0 are the non-positive residuals. One solution to achieve this could be to estimate an average production function by ordinary least squares and then simply shift the production function up until all residuals are negative or zero (see right panel of figure 6.1). However, this

244

6 Stochastic Frontier Analysis procedure does not account for statistical noise and is very sensitive to positive outliers.1 As virtually all data sets and models are flawed with statistical noise, e.g. due to measurement errors, omitted variables, and approximation errors, Meeusen and van den Broeck (1977) and Aigner, Lovell, and Schmidt (1977) independently proposed the stochastic frontier model that simultaneously accounts for statistical noise and technical inefficiency: ln y = ln f (x) − u + v

with u ≥ 0,

(6.2)

where −u ≤ 0 accounts for technical inefficiency and v accounts for statistical noise. This model can be re-written (see, e.g. Coelli et al., 2005, p. 243): y = f (x) e−u ev

(6.3)

As explained in section 5.1, Shepard’s output-oriented technical efficiencies are defined as the ratio between the observed output and the frontier output (see also, e.g., Coelli et al., 2005, p. 244): TE =

y f (x) e−u ev = = e−u , f (x) ev f (x) ev

(6.4)

while Farrell’s output-oriented technical efficiencies are defined as the ratio between the frontier output and the observed output: TE =

f (x) ev f (x) ev = = eu . y f (x) e−u ev

(6.5)

As Shepard’s output-oriented technical efficiencies are much more frequently used than Farrell’s output-oriented technical efficiencies, we will continue using Shepard’s output-oriented technical efficiencies. The stochastic frontier model is usually estimated by an econometric maximum likelihood estimation, which requires distributional assumptions of the error terms. Most often, it is assumed that the noise term v follows a normal distribution with zero mean and constant variance σv2 , the inefficiency term u follows a positive half-normal distribution or a positive truncated normal distribution with constant scale parameter σu2 , and all vs and all us are independent: v ∼ N (0, σv2 )

(6.6)

u ∼ N + (µ, σu2 ),

(6.7)

where µ = 0 for a positive half-normal distribution and µ 6= 0 for a positive truncated normal distribution. These assumptions result in a left-skewed distribution of the total error terms ε = −u + v, i.e. the density function is flat on the left and steep on the right. Hence, it is very rare that a firm has a large positive residual (much higher output than the production function) but 1

This is also true for the frequently-used Data Envelopment Analysis (DEA).

245

6 Stochastic Frontier Analysis it is not so rare that a firm has a large negative residual (much lower output than the production function). 6.1.1.1 Marginal products and output elasticities in SFA models Given the multiplicative specification of stochastic production frontier models (6.3) and assuming that the random error v is zero, we can see that the marginal products are downscaled by the level of the technical efficiency: ∂y f (x) ∂f (x) −u ∂f (x) = e = TE = T E αi ∂xi ∂xi ∂xi xi

(6.8)

However, the partial production elasticities are unaffected by the efficiency level: ∂y xi ∂f (x) xi ∂f (x) −u xi ∂ ln f (x) e = = αi = = −u ∂xi y ∂xi f (x)e ∂xi f (x) ∂ ln xi

(6.9)

As the output elasticities do not depend on the firm’s technical efficiency, also the elasticity of scale does not depend on the firm’s technical efficiency.

6.1.2 Skewness of residuals from OLS estimations The following commands plot histograms of the residuals taken from the Cobb-Douglas and the Translog production function: > hist( residuals( prodCD ), 15 )

10 15 20 0

5

Frequency

10 15 20 5 0

Frequency

> hist( residuals( prodTL ), 15 )

−1.5

−0.5

0.5 1.0 1.5

−1.5

residuals prodCD

−0.5

0.5 1.0 1.5

residuals prodTL

Figure 6.2: Residuals of Cobb-Douglas and Translog production functions The resulting graphs are shown in figure 6.2. The residuals of both production functions are left-skewed. This visual assessment of the skewness can be confirmed by calculating the skewness using the function skewness that is available in the package moments:

246

6 Stochastic Frontier Analysis > library( "moments" ) > skewness( residuals( prodCD ) ) [1] -0.4191323 > skewness( residuals( prodTL ) ) [1] -0.3194211 As a negative skewness means that the residuals are left-skewed, it is likely that not all apple producers are fully technically efficient. However, the distribution of the residuals does not always have the expected skewness. Possible reasons for an unexpected skewness of OLS residuals are explained in section 6.2.2.

6.1.3 Cobb-Douglas stochastic production frontier 6.1.3.1 Estimation We can use the command sfa (package frontier) to estimate stochastic production frontiers. The basic syntax of the command sfa is similar to the syntax of the command lm. The following command estimates a Cobb-Douglas stochastic production frontier assuming that the inefficiency term u follows a positive halfnormal distribution: > library( "frontier" ) > prodCDSfa summary( prodCDSfa ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 12 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept) 0.228813

1.247739

0.1834 0.8544981

log(qCap)

0.160934

0.081883

1.9654 0.0493668 *

log(qLab)

0.684777

0.146797

4.6648 3.089e-06 ***

log(qMat)

0.465871

0.131588

3.5404 0.0003996 ***

sigmaSq

1.000040

0.202456

4.9395 7.830e-07 ***

gamma

0.896664

0.070952 12.6375 < 2.2e-16 ***

247

6 Stochastic Frontier Analysis --Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -133.8893 cross-sectional data total number of observations = 140 mean efficiency: 0.5379937 The parameters of the Cobb-Douglas production frontier can be interpreted as before. The estimated production function is monotonically increasing in all inputs. The output elasticity of capital is 0.161, the output elasticity of labor is 0.685, The output elasticity of materials is 0.466, and the elasticity of scale is 1.312. 6.1.3.2 Variance parameters The estimation algorithm re-parameterizes the variance parameter of the noise term (σv2 ) and the scale parameter of the inefficiency term (σu2 ) and instead estimates the parameters σ 2 = σv2 + σu2 and γ = σu2 /σ 2 . The parameter γ lies between zero and one and indicates the importance of the inefficiency term. If γ is zero, the inefficiency term u is irrelevant and the results should be equal to OLS results. In contrast, if γ is one, the noise term v is irrelevant and all deviations from the production frontier are explained by technical inefficiency. As the estimate of γ is 0.897, we can conclude that both statistical noise and inefficiency are important for explaining deviations from the production function but that inefficiency is more important than noise. As σu2 is not equal to the variance of the inefficiency term u, the estimated parameter γ cannot be interpreted as the proportion of the total variance that is due to inefficiency. In fact, the variance of the inefficiency term u is



V ar(u) = σu2 1 − 

µ σu

φ

Φ





µ σu

µ σu







φ



µ σu

 2 

 −     , µ Φ σu

(6.10)

where Φ(.) indicates the cumulative distribution function and φ(.) the probability density function of the standard normal distribution. If the inefficiency term u follows a positive halfnormal distribution (i.e. µ = 0), the above equation reduces to h

i

V ar(u) = σu2 1 − (2 φ (0))2 ,

(6.11)

We can calculate the estimated variances of the inefficiency term u and the noise term v by following commands: > gamma sigmaSq sigmaSqU varU varV summary( prodCDSfa, extraPar = TRUE ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 12 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept) 0.228813

1.247739

0.1834 0.8544981

log(qCap)

0.160934

0.081883

1.9654 0.0493668 *

log(qLab)

0.684777

0.146797

4.6648 3.089e-06 ***

log(qMat)

0.465871

0.131588

3.5404 0.0003996 ***

2

This equation relies on the assumption that the inefficiency term u and the noise term v are independent, i.e. their covariance is zero.

249

6 Stochastic Frontier Analysis sigmaSq

1.000040

0.202456

4.9395 7.830e-07 ***

gamma

0.896664

0.070952 12.6375 < 2.2e-16 ***

sigmaSqU

0.896700

0.241715

3.7097 0.0002075 ***

sigmaSqV

0.103340

0.055831

1.8509 0.0641777 .

sigma

1.000020

0.101226

9.8791 < 2.2e-16 ***

sigmaU

0.946942

0.127629

7.4195 1.176e-13 ***

sigmaV

0.321465

0.086838

3.7019 0.0002140 ***

lambdaSq

8.677179

6.644542

1.3059 0.1915829

lambda

2.945705

1.127836

2.6118 0.0090061 **

varU

0.325843

NA

NA

NA

sdU

0.570827

NA

NA

NA

gammaVar

0.759217

NA

NA

NA

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -133.8893 cross-sectional data total number of observations = 140 mean efficiency: 0.5379937 The additionally returned parameter are defined as follows: sigmaSqU = σu2 = σ 2 · γ, sigmaSqV √ p p = σv2 = σ 2 · (1 − γ) = V ar (v), sigma = σ = σ 2 , sigmaU = σu = σu2 , sigmaV = σv = σv2 , lambdaSq = λ2 = σu2 /σv2 , lambda = λ = σu /σv , varU = V ar (u), sdU =

p

V ar (u), and gammaVar

= V ar (u)/(V ar (u) + V ar (v)). 6.1.3.3 Statistical tests for inefficiencies If there would be no inefficiencies, i.e. u = 0 for all observations, coefficient γ would be equal to zero. Hence, one can test the null hypothesis of no inefficiencies by simply testing whether γ is equal to (does not significantly deviate from) zero. However, a t-test of the null hypothesis γ = 0 (e.g. reported in the output of the summary method) is not valid, because γ is bound to the interval [0, 1] and hence, cannot follow a t-distribution. Instead, we can use a likelihood ratio test to check whether adding the inefficiency term u significantly improves the fit of the model. If the lrtest method is called just with a single stochastic frontier model, it compares the stochastic frontier model with the corresponding OLS model (i.e. a model with γ equal to zero): > lrtest( prodCDSfa ) Likelihood ratio test

250

6 Stochastic Frontier Analysis Model 1: OLS (no inefficiency) Model 2: Error Components Frontier (ECF) #Df

LogLik Df

1

5 -137.61

2

6 -133.89

Chisq Pr(>Chisq)

1 7.4387

0.003192 **

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Under the null hypothesis (no inefficiency, only noise), the test statistic asymptotically follows a mixed χ2 -distribution (Coelli, 1995).3 The rather small P-value indicates that the data clearly reject the OLS model in favor of the stochastic frontier model, i.e. there is significant technical inefficiency. 6.1.3.4 Obtaining technical efficiency estimates As neither the noise term v nor the inefficiency term u but only the total error term ε = −u + v is known, the technical efficiencies T E = e−u are generally unknown. However, given that the parameter estimates (including the parameters σ 2 and γ or σv2 and σu2 ) and the total error term ε are known, it is possible to determine the expected value of the technical efficiency (see, e.g. Coelli et al., 2005, p. 255): Td E = E e−u 



(6.12)

These efficiency estimates can be obtained by the efficiencies method: > dat$effCD plot( residuals( prodCDSfa ), dat$effCD ) > hist( dat$effCD, 15 ) The resulting graphs are shown in figure 6.3. The efficiency estimates are rather low: the firms only produce between 10% and 90% of the maximum possible output quantities. We explore the correlation between firm size (measured as output quantity as well as aggregate input quantity indicated by a Fisher quantity index of all inputs) and the efficiency estimates: > plot( dat$qOut, dat$effCD, log = "x" ) > plot( dat$X, dat$effCD, log = "x" ) 3

As a standard likelihood ratio test assumes that the test statistic follows a (standard) χ2 -distribution under the null hypothesis, a test that is conducted by the command lrtest( prodCD, prodCDSfa ) returns an incorrect P-value.

251

15 0

0.2

5

10

Frequency

0.6 0.4

effCD

0.8

6 Stochastic Frontier Analysis

−2.5

−1.5

−0.5

0.5

0.2

residCD

0.4

0.6

0.8

effCD

0.8 0.6 0.2

0.4

effCD

0.6 0.4 0.2

effCD

0.8

Figure 6.3: Efficiency estimates of Cobb-Douglas production frontier

1e+05

5e+05

5e+06

0.5

qOut

1.0

2.0

X

Figure 6.4: Firm size and efficiency estimates of Cobb-Douglas production frontier

252

5.0

6 Stochastic Frontier Analysis The resulting graphs are shown in figure 6.4. As the efficiency directly influences the output quantity, it is not surprising that the efficiency estimates are highly correlated with the output quantity. On the other hand, the efficiency estimates are only slightly correlated with firm size measured as aggregate input quantity. However, the largest firms all have an above-average efficiency estimate. 6.1.3.5 Truncated normal distribution of the inefficiency term Instead of assuming that the inefficiency term u follows a half-normal distribution, we can assume that it follows a truncated norm distribution: u ∼ N + (µ, σu2 )

(6.13)

If the location parameter µ is equal to zero, the truncated normal distribution is identical to the half-normal distribution. If argument truncNorm of function sfa() of the frontier package is set to TRUE, this function estimates a stochastic frontier model assuming a truncated normal distribution of the inefficiency term: > prodCDSfaTn summary( prodCDSfaTn, extraPar = TRUE ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 13 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept) 0.227323

1.250360

0.1818 0.8557349

log(qCap)

0.160632

0.079397

2.0232 0.0430568 *

log(qLab)

0.688275

0.148699

4.6286 3.681e-06 ***

log(qMat)

0.464421

0.132878

3.4951 0.0004739 ***

sigmaSq

0.934405

0.599621

1.5583 0.1191562

gamma

0.895931

0.069525 12.8865 < 2.2e-16 ***

mu

0.130129

1.145962

0.1136 0.9095912

sigmaSqU

0.837163

0.557136

1.5026 0.1329373

sigmaSqV

0.097242

0.077929

1.2478 0.2120920

253

6 Stochastic Frontier Analysis sigma

0.966646

0.310155

3.1167 0.0018292 **

sigmaU

0.914966

0.304457

3.0052 0.0026537 **

sigmaV

0.311837

0.124951

2.4957 0.0125720 *

lambdaSq

8.609044

6.419490

1.3411 0.1798948

lambda

2.934117

1.093939

2.6822 0.0073149 **

varU

0.331133

NA

NA

NA

sdU

0.575442

NA

NA

NA

gammaVar

0.772998

NA

NA

NA

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -133.8836 cross-sectional data total number of observations = 140 mean efficiency: 0.5272536 The results still indicate statistical significant technical inefficiency: > lrtest( prodCDSfaTn ) Likelihood ratio test Model 1: OLS (no inefficiency) Model 2: Error Components Frontier (ECF) #Df

LogLik Df Chisq Pr(>Chisq)

1

5 -137.61

2

7 -133.88

2

7.45

0.0092 **

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

However, neither the t-test presented in the summary output above nor a likelihood ratio test can reject the model with the half-normal distribution of the inefficiency term (i.e., with µ = 0) in favour of the model with truncated normal distribution of the inefficiency term: > lrtest( prodCDSfa, prodCDSfaTn ) Likelihood ratio test Model 1: prodCDSfa Model 2: prodCDSfaTn #Df

LogLik Df

Chisq Pr(>Chisq)

254

6 Stochastic Frontier Analysis 1

6 -133.89

2

7 -133.88

1 0.0113

0.9153

Given the high P-values of these two tests, it seems to be reasonable to use the model with the half-normal distribution of the error term.

6.1.4 Translog production frontier 6.1.4.1 Estimation As the Cobb-Douglas functional form is very restrictive, we additionally estimate a Translog stochastic production frontier: > prodTLSfa summary( prodTLSfa, extraPar = TRUE ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 23 iterations: cannot find a parameter vector that results in a log-likelihood value larger than the log-likelihood value obtained in the previous step final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept)

-8.8110424 19.9181625 -0.4424 0.6582271

log(qCap)

-0.6332521

log(qLab)

4.4511064

log(qMat)

-1.3976309

2.0855273 -0.3036 0.7614012 4.4552359

0.9991 0.3177593

3.8097809 -0.3669 0.7137284

I(0.5 * log(qCap)^2)

0.0053258

I(0.5 * log(qLab)^2)

-1.5030433

0.6812813 -2.2062 0.0273700 *

I(0.5 * log(qMat)^2)

-0.5113559

0.3733348 -1.3697 0.1707812

I(log(qCap) * log(qLab))

0.4187529

0.1866174

0.2747251

0.0285 0.9772324

1.5243 0.1274434

I(log(qCap) * log(qMat)) -0.4371561

0.1902856 -2.2974 0.0215978 *

I(log(qLab) * log(qMat))

0.9800294

0.4216638

2.3242 0.0201150 *

sigmaSq

0.9587307

0.1968009

4.8716 1.107e-06 ***

gamma

0.9153387

0.0647478 14.1370 < 2.2e-16 ***

sigmaSqU

0.8775633

0.2328364

255

3.7690 0.0001639 ***

6 Stochastic Frontier Analysis sigmaSqV

0.0811674

0.0497448

1.6317 0.1027476

sigma

0.9791480

0.1004960

9.7432 < 2.2e-16 ***

sigmaU

0.9367835

0.1242744

7.5380 4.771e-14 ***

sigmaV

0.2848989

0.0873025

3.2634 0.0011010 **

10.8117752

9.0334826

1.1969 0.2313628

lambda

3.2881264

1.3736520

2.3937 0.0166789 *

varU

0.3188892

NA

NA

NA

sdU

0.5647027

NA

NA

NA

gammaVar

0.7971103

NA

NA

NA

lambdaSq

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -128.0684 cross-sectional data total number of observations = 140 mean efficiency: 0.5379939 6.1.4.2 Statistical test for inefficiencies A likelihood ratio test confirms that the stochastic frontier model fits the data much better than an average production function estimated by OLS: > lrtest( prodTLSfa ) Likelihood ratio test Model 1: OLS (no inefficiency) Model 2: Error Components Frontier (ECF) #Df

LogLik Df Chisq Pr(>Chisq)

1

11 -131.25

2

12 -128.07

1 6.353

0.005859 **

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

6.1.4.3 Testing against the Cobb-Douglas functional form A further likelihood ratio test indicates that it is not really clear whether the Translog stochastic frontier model fits the data significantly better than the Cobb-Douglas stochastic frontier model: > lrtest( prodCDSfa, prodTLSfa )

256

6 Stochastic Frontier Analysis Likelihood ratio test Model 1: prodCDSfa Model 2: prodTLSfa #Df

LogLik Df

1

6 -133.89

2

12 -128.07

Chisq Pr(>Chisq)

6 11.642

0.07045 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

While the Cobb-Douglas functional form is accepted at the 5% significance level, it is rejected in favor of the Translog functional form at the 10% significance level. 6.1.4.4 Obtaining technical efficiency estimates The efficiency estimates based on the Translog stochastic production frontier can be obtained (again) by the efficiencies method: > dat$effTL hist( dat$effTL, 15 ) > plot( dat$qOut, dat$effTL, log = "x" )

0.8 0.6 0.4

effTL

0.6 0.4

effTL

8 6

0.2

0

2

0.2

4

Frequency

10

0.8

12

> plot( dat$X, dat$effTL, log = "x" )

0.2

0.4

0.6

0.8

1e+05

5e+05

effTL

5e+06

0.5

qOut

1.0

2.0

5.0

X

Figure 6.5: Efficiency estimates of Translog production frontier The resulting graphs are shown in figure 6.5. These efficiency estimates are rather similar to the efficiency estimates based on the Cobb-Douglas stochastic production frontier. This is confirmed by a direct comparison of these efficiency estimates:

257

0.2

0.4

effTL

0.6

0.8

6 Stochastic Frontier Analysis

0.2

0.4

0.6

0.8

effCD

Figure 6.6: Efficiency estimates of Cobb-Douglas and Translog production frontier > compPlot( dat$effCD, dat$effTL ) The resulting graph is shown in figure 6.6. Most efficiency estimates only slightly differ between the two functional forms but a few efficiency estimates are considerably higher for the Translog functional form. The inflexibility of the Cobb-Douglas functional form probably resulted in an insufficient adaptation of the frontier to some observations, which lead to larger negative residuals and hence, lower efficiency estimates in the Cobb-Douglas model. 6.1.4.5 Truncated normal distribution of the inefficiency term As explained in section 6.1.3.5, we can estimate a stochastic frontier model based on the assumption that the inefficiency term u follows a truncated normal distribution instead of a half-normal distribution: > prodTLSfaTn summary( prodTLSfaTn, extraPar = TRUE ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 30 iterations:

258

6 Stochastic Frontier Analysis log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value (Intercept)

Pr(>|z|)

-15.759268

21.912418 -0.7192

0.472022

log(qCap)

-0.748266

2.157070 -0.3469

0.728674

log(qLab)

5.349481

4.806460

1.1130

0.265718

log(qMat)

-0.979857

3.971830 -0.2467

0.805139

I(0.5 * log(qCap)^2)

-0.050523

0.184108 -0.2744

0.783760

I(0.5 * log(qLab)^2)

-1.612095

0.723666 -2.2277

0.025902 *

I(0.5 * log(qMat)^2)

-0.552201

0.399247 -1.3831

0.166633

0.287449

0.100636

I(log(qCap) * log(qLab))

0.471927

I(log(qCap) * log(qMat))

-0.428907

I(log(qLab) * log(qMat))

1.6418

0.199232 -2.1528

0.031334 *

0.969984

0.448343

2.1635

0.030504 *

sigmaSq

0.644764

0.221585

2.9098

0.003617 **

gamma

0.939671

0.067754 13.8689 < 2.2e-16 ***

mu

0.658349

0.413449

1.5923

0.111310

sigmaSqU

0.605866

0.213788

2.8340

0.004598 **

sigmaSqV

0.038898

0.045373

0.8573

0.391281

sigma

0.802972

0.137978

5.8196 5.900e-09 ***

sigmaU

0.778374

0.137330

5.6679 1.445e-08 ***

sigmaV

0.197225

0.115028

1.7146

0.086420 .

15.575816

18.615851

0.8367

0.402763

lambda

3.946621

2.358454

1.6734

0.094250 .

varU

0.353965

NA

NA

NA

sdU

0.594950

NA

NA

NA

gammaVar

0.900989

NA

NA

NA

lambdaSq

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -127.581 cross-sectional data total number of observations = 140 mean efficiency: 0.4613383 The fit of the Translog stochastic production frontier with truncated normal inefficiency term is significantly better than the fit of the Cobb-Douglas stochastic production frontier with truncated normal inefficiency term (at 5% significance level):

259

6 Stochastic Frontier Analysis > lrtest( prodCDSfaTn, prodTLSfaTn ) Likelihood ratio test Model 1: prodCDSfaTn Model 2: prodTLSfaTn #Df

LogLik Df

1

7 -133.88

2

13 -127.58

Chisq Pr(>Chisq)

6 12.605

0.04975 *

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The results indicate statistical significant technical inefficiency: > lrtest( prodTLSfaTn ) Likelihood ratio test Model 1: OLS (no inefficiency) Model 2: Error Components Frontier (ECF) #Df

LogLik Df

1

11 -131.25

2

13 -127.58

Chisq Pr(>Chisq)

2 7.3279

0.009803 **

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

However, neither the t-test presented in the summary output above nor a likelihood ratio test can reject the model with the half-normal distribution of the inefficiency term (i.e., with µ = 0) in favour of the model with truncated normal distribution of the inefficiency term (at 10% significance level): > lrtest( prodTLSfa, prodTLSfaTn ) Likelihood ratio test Model 1: prodTLSfa Model 2: prodTLSfaTn #Df

LogLik Df

1

12 -128.07

2

13 -127.58

Chisq Pr(>Chisq)

1 0.9748

0.3235

260

6 Stochastic Frontier Analysis

6.1.5 Translog production frontier with mean-scaled variables As argued in section 2.6.14, it is sometimes convenient to estimate a Translog production (frontier) function with mean-scaled variables. The following command estimates a Translog production function with mean-scaled output and input quantities: > prodTLmSfa summary( prodTLmSfa ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 17 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept)

0.6388793

0.1311531

4.8712 1.109e-06 ***

log(qmCap)

0.1308903

0.1003318

1.3046

log(qmLab)

0.7065404

0.1555606

4.5419 5.575e-06 ***

log(qmMat)

0.4657266

0.1516483

3.0711

0.002133 **

I(0.5 * log(qmCap)^2)

0.0053227

0.1848995

0.0288

0.977034

I(0.5 * log(qmLab)^2)

-1.5030266

0.6761522 -2.2229

0.026222 *

I(0.5 * log(qmMat)^2)

-0.5113617

0.3749803 -1.3637

0.172661

0.2686428

0.119047

I(log(qmCap) * log(qmLab))

0.4187571

1.5588

0.192038

I(log(qmCap) * log(qmMat)) -0.4371473

0.1886950 -2.3167

0.020521 *

I(log(qmLab) * log(qmMat))

0.9800162

0.4201674

2.3324

0.019677 *

sigmaSq

0.9587158

0.1967744

4.8722 1.104e-06 ***

gamma

0.9153349

0.0659588 13.8774 < 2.2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -128.0684 cross-sectional data total number of observations = 140

261

6 Stochastic Frontier Analysis mean efficiency: 0.5379969 > all.equal( coef( prodTLmSfa )[-c(1:4)], coef( prodTLSfa )[-c(1:4)], +

check.attributes = FALSE )

[1] "Mean relative difference: 1.229993e-05" > all.equal( efficiencies( prodTLmSfa ), efficiencies( prodTLSfa ) ) [1] "Mean relative difference: 7.059797e-06" While the intercept and the first-order parameters have adjusted to the new units of measurement, the second-order parameters, the variance parameters, and the efficiency estimates remain (nearly) unchanged. From the estimated coefficients of the Translog production frontier with mean-scaled input quantities, we can immediately see that the monotonicity condition is fulfilled at the sample mean, that the output elasticities of capital, labor, and materials are 0.131, 0.707, and 0.466, respectively, at the sample mean, and that the elasticity of scale is 0.131 + 0.707 + 0.466 = 1.303 at the sample mean.

6.2 Stochastic cost frontiers 6.2.1 Specification The general specification of a stochastic cost frontier is ln c = ln c(w, y) + u + v

with u ≥ 0,

(6.14)

where u ≥ 0 accounts for cost inefficiency and v accounts for statistical noise. This model can be re-written as: c = c(w, y) eu ev

(6.15)

The cost efficiency according to Shepard is CE =

c f (x) eu ev = = eu , c(w, y) ev c(w, y) ev

(6.16)

while the cost efficiency according to Farrell is CE =

c(w, y) ev c(w, y) ev = = e−u . c f (x) eu ev

(6.17)

Assuming a normal distribution of the noise term v and a positive half-normal distribution of the inefficiency term u, the distribution of the residuals from a cost function is expected to be right-skewed in the case of cost inefficiencies.

262

6 Stochastic Frontier Analysis

6.2.2 Skewness of residuals from OLS estimations The following commands visualize the distribution of the residuals of the OLS estimations of the Cobb-Douglas and Translog cost functions with linear homogeneity in input prices imposed: > hist( residuals( costCDHom ) )

0

10 20 30 40

Frequency

15 0 5

Frequency

25

> hist( residuals( costTLHom ) )

−0.5

0.0

0.5

−0.5

residuals costCDHom

0.0

0.5

1.0

residuals costTLHom

Figure 6.7: Residuals of Cobb-Douglas and Translog cost functions The resulting graphs are shown in figure 6.7. The distributions of the residuals look approximately symmetric and rather a little left-skewed than right-skewed (although we expected the latter). This visual assessment of the skewness can be confirmed by calculating the skewness using the function skewness that is available in the package moments: > library( "moments" ) > skewness( residuals( costCDHom ) ) [1] -0.05788105 > skewness( residuals( costTLHom ) ) [1] -0.03709506 The residuals of the two cost functions have both a small (in absolute terms) but negative skewness, which means that the residuals are slightly left-skewed, although we expected rightskewed residuals. It could be that the distribution of the unknown true total error term (u + v) in the sample is indeed symmetric or slightly left-skewed, e.g. because ˆ there is no cost inefficiency (but only noise) (the distribution of residuals is “correct”), ˆ the distribution of the noise term is left-skewed, which neutralizes the right-skewed distri-

bution of the inefficiency term (misspecification of the distribution of the noise term in the SFA model),

263

6 Stochastic Frontier Analysis ˆ the distribution of the inefficiency term is symmetric or left-skewed (misspecification of the

distribution of the inefficiency term in the SFA model), ˆ the sampling of the observations by coincidence resulted in a symmetric or left-skewed distribution of the true total error term (u+v) in this specific sample, although the distribution of the true total error term (u + v) in the population is right-skewed, and/or ˆ the farm managers do not aim at maximizing profit (which implies minimizing costs) but have other objectives. It could also be that the distribution of the unknown true residuals in the sample is right-skewed, but the OLS estimates are left-skewed, e.g. because ˆ the parameter estimates are imprecise (but unbiased), ˆ the estimated functional forms (Cobb-Douglas and Translog) are poor approximations of

the unknown true functional form (functional-form misspecification), ˆ there are further relevant explanatory variables that are not included in the model specification (omitted-variables bias), ˆ there are measurement errors in the variables, particularly in the explanatory variables (errors-in-variables problem), and/or ˆ the output quantity or the input prices are not exogenously given (endogeneity bias). Hence, a left-skewed distribution of the residuals does not necessarily mean that there is no cost inefficiency, but it could also mean that the model is misspecified or that this is just by coincidence.

6.2.3 Estimation of a Cobb-Douglas stochastic cost frontier The following command estimates a Cobb-Douglas stochastic cost frontier with linear homogeneity in input prices imposed: > costCDHomSfa summary( costCDHomSfa ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency increases the endogenous variable (as in a cost function) The dependent variable is logged Iterative ML estimation terminated after 57 iterations: cannot find a parameter vector that results in a log-likelihood value larger than the log-likelihood value obtained in the previous step final maximum likelihood estimates Estimate Std. Error z value

264

Pr(>|z|)

6 Stochastic Frontier Analysis (Intercept)

6.74954369 0.67670662

9.9741 < 2.2e-16 ***

log(pCap/pMat) 0.07242021 0.04357862

1.6618

0.09655 .

log(pLab/pMat) 0.44641208 0.07762778

5.7507 8.889e-09 ***

log(qOut)

0.37415805 0.02823500 13.2516 < 2.2e-16 ***

sigmaSq

0.11117238 0.01439869

7.7210 1.154e-14 ***

gamma

0.00015081 0.06235873

0.0024

0.99807

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -44.87812 cross-sectional data total number of observations = 140 mean efficiency: 0.9967413 The parameter γ, which indicates the proportion of the total residual variance that is caused by inefficiency is close to zero and a t-test suggests that it is statistically not significantly different from zero. As the t-test for the parameter γ is not always reliable, we use a likelihood ratio test to verify this result: > lrtest( costCDHomSfa ) Likelihood ratio test Model 1: OLS (no inefficiency) Model 2: Error Components Frontier (ECF) #Df

LogLik Df Chisq Pr(>Chisq)

1

5 -44.878

2

6 -44.878

1

0

0.4991

This test confirms that the fit of the OLS model (which assumes that γ is zero and hence, that there is no inefficiency) is not significantly worse than the fit of the stochastic frontier model. In fact, the cost efficiency estimates are all very close to one. By default, the efficiencies() method calculates the efficiency estimates as E [e−u ], which means that we obtain estimates of Farrell-type cost efficiencies (6.17). Given that E [eu ] is not equal to 1/E [e−u ] (as the expectation operator is an additive operator), we cannot obtain estimates of Shepard-type cost efficiencies (6.16) by taking the inverse of the estimates of the Farrell-type cost efficiencies (6.17). However, we can obtain estimates of Shepard-type cost efficiencies (6.16) by setting argument minusU of the efficiencies() method equal to FALSE, which tells the efficiencies() method to calculate the efficiency estimates as E [eu ].

265

6 Stochastic Frontier Analysis > dat$costEffCDHomFarrell dat$costEffCDHomShepard hist( dat$costEffCDHomFarrell, 15 )

15 10 0

5

Frequency

15 10 5 0

Frequency

> hist( dat$costEffCDHomShepard, 15 )

0.99670 0.99672 0.99674 0.99676 0.99678

1.00324

costEffCDHomFarrell

1.00328

1.00332

costEffCDHomShepard

Figure 6.8: Efficiency estimates of Cobb-Douglas cost frontier The resulting graphs are shown in figure 6.8. While the Farrell-type cost efficiencies are all slightly below one, the Shepard-type cost efficiencies are all slightly above one. Both graphs show that we do not find any relevant cost inefficiencies, although we have found considerable technical inefficiencies.

6.2.4 Short-run cost frontiers

20 10 0

Frequency

30

> hist( residuals( costCDSRHom ) )

−0.5

0.0

0.5

residuals costCDSRHom

Figure 6.9: Residuals of Cobb-Douglas short-run cost function The resulting graph is shown in figure 6.9. > costCDSRHomSfa summary( costCDSRHomSfa ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency increases the endogenous variable (as in a cost function) The dependent variable is logged

266

6 Stochastic Frontier Analysis Iterative ML estimation terminated after 58 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value (Intercept)

Pr(>|z|)

5.6697690

1.0771388

5.2637 1.412e-07 ***

log(pLab/pMat) 0.5348814

0.0609301

8.7786 < 2.2e-16 ***

log(qCap)

0.1877461

0.0376363

4.9884 6.087e-07 ***

log(qOut)

0.2900899

0.0279995 10.3605 < 2.2e-16 ***

sigmaSq

0.0980818

0.0215985

4.5411 5.595e-06 ***

gamma

0.0013434

0.2708684

0.0050

0.996

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -36.05537 cross-sectional data total number of observations = 140 mean efficiency: 0.9909068 > costCDSRHomSfa summary( costCDSRHomSfa ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency increases the endogenous variable (as in a cost function) The dependent variable is logged Iterative ML estimation terminated after 41 iterations: cannot find a parameter vector that results in a log-likelihood value larger than the log-likelihood value obtained in the previous step final maximum likelihood estimates Estimate Std. Error z value (Intercept)

Pr(>|z|)

5.6700012

0.9773806

5.8012 6.583e-09 ***

log(pLab/pMat) 0.5348666

0.0662204

8.0771 6.634e-16 ***

log(qCap)

0.1877402

0.0381704

4.9185 8.722e-07 ***

log(qOut)

0.2900985

0.0312842

9.2730 < 2.2e-16 ***

sigmaSq

0.0980771

0.0196235

4.9980 5.794e-07 ***

267

6 Stochastic Frontier Analysis gamma

0.0012506

0.2555038

0.0049

0.9961

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -36.05537 cross-sectional data total number of observations = 140 mean efficiency: 0.9912245 > dat$costEffCDSRHom hist( dat$costEffCDSRHom, 15 )

0.9910

0.9912

0.9914

0.9916

costEffCDSRHom

Figure 6.10: Efficiency estimates of Cobb-Douglas short-run cost frontier The resulting graphs are shown in figure 6.10.

6.2.5 Profit frontiers ln π = ln π(p, w) − u + v

with u ≥ 0,

(6.18)

where −u ≤ 0 accounts for profit inefficiency and v accounts for statistical noise. This model can be re-written as: π = π(p, w) e−u ev

(6.19)

Profit efficiency according to Farrell: PE =

π(w, y) e−u ev π = = e−u π(w, y) ev π(w, y) ev

(6.20)

6.3 Analyzing the effects of z variables In many empirical cases, the output quantity does not only depend on the input quantities but also on some other variables, e.g. the manager’s education and experience and in agricultural production also the soil quality and rainfall. If these factors influence the production process,

268

6 Stochastic Frontier Analysis they must be included in applied production analyses in order to avoid an omitted-variables bias. Our data set on French apple producers includes the variable adv, which is a dummy variable and indicates whether the apple producer uses an advisory service. In the following, we will apply different methods to figure out whether the production process differs between users and non-users of an advisory service.

6.3.1 Production functions with z variables Additional factors that influence the production process (z) can be included as additional explanatory variables in the production function: y = f (x, z).

(6.21)

This function can be used to analyze how the additional explanatory variables (z) affect the output quantity for given input quantities, i.e. how they affect the productivity. In case of a Cobb-Douglas functional form, we get following extended production function: ln y = α0 +

X

αi ln xi + αz z

(6.22)

i

Based on this Cobb-Douglas production function and our data set on French apple producers, we can check whether the apple producers who use an advisory service produce a different output quantity than non-users with the same input quantities, i.e. whether the productivity differs between users and non-users. This extended production function can be estimated by following command: > prodCDAdv summary( prodCDAdv ) Call: lm(formula = log(qOut) ~ log(qCap) + log(qLab) + log(qMat) + adv, data = dat) Residuals: Min

1Q

Median

3Q

Max

-1.7807 -0.3821

0.0022

0.4709

1.3323

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -2.33371

1.29590

-1.801

0.0740 .

log(qCap)

0.08581

1.826

0.0700 .

0.15673

269

6 Stochastic Frontier Analysis log(qLab)

0.69225

0.15190

4.557 1.15e-05 ***

log(qMat)

0.62814

0.12379

5.074 1.26e-06 ***

adv

0.25896

0.10932

2.369

0.0193 *

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.6452 on 135 degrees of freedom Multiple R-squared: F-statistic:

0.6105,

Adjusted R-squared:

52.9 on 4 and 135 DF,

p-value: < 2.2e-16

0.599

The estimation result shows that users of an advisory service produce significantly more than non-users with the same input quantities. Given the Cobb-Douglas production function (6.22), the coefficient of an additional explanatory variable can be interpreted as the marginal effect on the relative change of the output quantity: αz =

∂ ln y ∂ ln y ∂y ∂y 1 = = ∂z ∂y ∂z ∂z y

(6.23)

Hence, our estimation result indicates that users of an advisory service produce approximately 25.9% more output than non-users with the same input quantity but the large standard error of this coefficient indicates that this estimate is rather imprecise. Given that the change of a dummy variable from zero to one is not marginal and that the coefficient of the variable adv is not close to zero, the above interpretation of this coefficient is a rather poor approximation. In fact, our estimation results suggest that the output quantity of apple producers with advisory service is on average exp(αz ) = 1.296 times as large as (29.6% larger than) the output quantity of apple producers without advisory service given the same input quantities. As users and non-users of an advisory service probably differ in some unobserved variables that affect the productivity (e.g. motivation and effort to increase productivity), the coefficient az is not necessarily the causal effect of the advisory service but describes the difference in productivity between users and non-users of the advisory service.

6.3.2 Production frontiers with z variables A production function that includes additional factors that influence the production process (6.21) can also be estimated as a stochastic production frontier. In this specification, it is assumed that the additional explanatory variables influence the production frontier. The following command estimates the extended Cobb-Douglas production function (6.22) using the stochastic frontier method: > prodCDAdvSfa summary( prodCDAdvSfa )

270

6 Stochastic Frontier Analysis Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 14 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept) -0.247751

1.357917 -0.1824 0.8552300

log(qCap)

0.156906

0.081337

1.9291 0.0537222 .

log(qLab)

0.695977

0.148793

4.6775 2.904e-06 ***

log(qMat)

0.491840

0.139348

3.5296 0.0004162 ***

adv

0.150742

0.111233

1.3552 0.1753583

sigmaSq

0.916031

0.231604

3.9552 7.648e-05 ***

gamma

0.861029

0.114087

7.5471 4.450e-14 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -132.8679 cross-sectional data total number of observations = 140 mean efficiency: 0.5545099 The estimation result still indicates that users of an advisory service have a higher productivity than non users, but the coefficient is smaller and no longer statistically significant. The result of the t-test is confirmed by a likelihood-ratio test: > lrtest( prodCDSfa, prodCDAdvSfa ) Likelihood ratio test Model 1: prodCDSfa Model 2: prodCDAdvSfa #Df

LogLik Df

1

6 -133.89

2

7 -132.87

Chisq Pr(>Chisq)

1 2.0428

0.1529

The model with advisory service as additional explanatory variable indicates that there are significant inefficiencies (at 5% significance level):

271

6 Stochastic Frontier Analysis > lrtest( prodCDAdvSfa ) Likelihood ratio test Model 1: OLS (no inefficiency) Model 2: Error Components Frontier (ECF) #Df

LogLik Df Chisq Pr(>Chisq)

1

6 -134.76

2

7 -132.87

1

3.78

0.02593 *

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The following commands compute the technical efficiency estimates and compare them to the efficiency estimates obtained from the Cobb-Douglas production frontier without advisory service as an explanatory variable: > dat$effCDAdv compPlot( dat$effCD[ dat$adv == 0 ], +

dat$effCDAdv[ dat$adv == 0 ] )

> points( dat$effCD[ dat$adv == 1 ], +

dat$effCDAdv[ dat$adv == 1 ], pch = 20 )

The resulting graph is shown in figure 6.11. It appears as if the non-users of an advisory service became somewhat more efficient. This is because the stochastic frontier model that includes the advisory service as an explanatory variable has in fact two production frontiers: a lower frontier for the non-users of an advisory service and a higher frontier for the users of an advisory service. The coefficient of the dummy variable adv, i.e. αadv , can be interpreted as a quick estimate of the difference between the two frontier functions. In our empirical case, the difference is approximately 15.1%. However, a precise calculation indicates that the frontier of the users of the advisory service is exp (αadv ) = 1.163 times (16.3% higher than) the frontier of the non-users of advisory service. And the frontier of the non-users of the advisory service is exp (−αadv ) = 0.86 times (14% lower than) the frontier of the users of advisory service. As the non-users of an advisory service are compared to a lower frontier now, they appear to be more efficient now. While it is reasonable to have different frontier functions for different soil types, it does not seem to be too reasonable to have different frontier functions for users and non-users of an advisory service, because there is no physical reasons, why users of an advisory service should have a maximum output quantity that is different from the maximum output quantity of non-users.

272

0.8 0.6 0.4 0.2

Production frontier with advisory service

6 Stochastic Frontier Analysis

0.2

0.4

0.6

0.8

Production frontier without advisory service

Figure 6.11: Technical efficiency estimates of Cobb-Douglas production frontier with and without advisory service as additional explanatory variable (circles = producers who do not use an advisory service, solid dots = producers who use an advisory service

6.3.3 Efficiency effects production frontiers As explained above, it does not seem to be too reasonable to have different frontier functions for users and non-users of an advisory service. However, it seems to be reasonable to assume that users of an advisory service have on average different efficiencies than non-users. A model that can account for this has been proposed by Battese and Coelli (1995). In this stochastic frontier model, the efficiency level might be affected by additional explanatory variables: The inefficiency term u follows a positive truncated normal distribution with constant scale parameter σu2 and a location parameter µ that depends on additional explanatory variables: u ∼ N + (µ, σu2 )

with µ = δ z,

(6.24)

where δ is an additional parameter (vector) to be estimated. Function sfa can also estimate these “efficiency effects frontiers”. The additional variables that should explain the efficiency level must be specified at the end of the model formula, where a vertical bar separates them from the (regular) input variables: > prodCDSfaAdvInt summary( prodCDSfaAdvInt ) Efficiency Effects Frontier (see Battese & Coelli 1995) Inefficiency decreases the endogenous variable (as in a production function)

273

6 Stochastic Frontier Analysis The dependent variable is logged Iterative ML estimation terminated after 18 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value (Intercept)

-0.090747

Pr(>|z|)

1.215594 -0.0747 0.9404914

log(qCap)

0.168625

0.082138

2.0529 0.0400775 *

log(qLab)

0.653868

0.142849

4.5773

log(qMat)

0.513527

0.132644

3.8715 0.0001082 ***

4.71e-06 ***

Z_(Intercept) -0.016714

1.220842 -0.0137 0.9890769

Z_adv

1.057873 -1.0186 0.3083926

-1.077550

sigmaSq

1.096441

0.747990

1.4659 0.1426891

gamma

0.863087

0.094468

9.1363 < 2.2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -130.516 cross-sectional data total number of observations = 140 mean efficiency: 0.600435 One can use the lrtest() method to test the statistical significance of the entire inefficiency model, i.e. the null hypothesis is H0 : γ = 0 and δj = 0 ∀ j: > lrtest( prodCDSfaAdvInt ) Likelihood ratio test Model 1: OLS (no inefficiency) Model 2: Efficiency Effects Frontier (EEF) #Df

LogLik Df

1

5 -137.61

2

8 -130.52

Chisq Pr(>Chisq)

3 14.185

0.001123 **

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The test indicates that the fit of this model is significantly better than the fit of the OLS model (without advisory service as explanatory variable).

274

6 Stochastic Frontier Analysis The coefficient of the advisory service in the inefficiency model is negative but statistically insignificant. By default, an intercept is added to the inefficiency model but it is completely statistically insignificant. In many econometric estimations of the efficiency effects frontier model, the intercept of the inefficiency model (δ0 ) is only weakly identified, because the values of δ0 can often be changed with only marginally reducing the log-likelihood value, if the slope parameters of the inefficiency model (δi , i 6= 0) and the variance parameters (σ 2 and γ) are adjusted accordingly. This can be checked by taking a look at the correlation matrix of the estimated parameters: > round( cov2cor( vcov( prodCDSfaAdvInt ) ), 2 ) (Intercept) log(qCap) log(qLab) log(qMat) Z_(Intercept) Z_adv (Intercept)

1.00

-0.06

-0.48

-0.21

0.07

0.08

log(qCap)

-0.06

1.00

-0.37

-0.17

log(qLab)

-0.48

-0.37

1.00

-0.57

log(qMat)

-0.21

-0.17

-0.57

1.00

Z_(Intercept)

0.07

-0.13

0.20

-0.11

1.00

0.91

Z_adv

0.08

-0.13

0.26

-0.20

0.91

1.00

sigmaSq

0.03

0.10

-0.14

0.00

-0.95 -0.89

gamma

0.29

-0.03

0.07

-0.34

-0.55 -0.46

-0.13 -0.13 0.20

0.26

-0.11 -0.20

sigmaSq gamma (Intercept)

0.03

log(qCap)

0.10 -0.03

log(qLab) log(qMat)

-0.14

0.29 0.07

0.00 -0.34

Z_(Intercept)

-0.95 -0.55

Z_adv

-0.89 -0.46

sigmaSq

1.00

0.73

gamma

0.73

1.00

The estimate of the intercept of the inefficiency model (δ0 ) is very highly correlated with the estimate of the (slope) coefficient of the advisory service in the inefficiency model (δ1 ) and the estimate of the parameter σ 2 and it is considerably correlated with the estimate of the parameter γ. The intercept can be suppressed by adding a “-1” to the specification of the inefficiency model: > prodCDSfaAdv summary( prodCDSfaAdv ) Efficiency Effects Frontier (see Battese & Coelli 1995) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged

275

6 Stochastic Frontier Analysis Iterative ML estimation terminated after 14 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept) -0.090455

1.247496 -0.0725

0.94220

log(qCap)

0.168471

0.077008

2.1877

0.02869 *

log(qLab)

0.654341

0.139669

4.6849 2.800e-06 ***

log(qMat)

0.513291

0.130854

3.9226 8.759e-05 ***

Z_adv

-1.064859

0.545950 -1.9505

0.05112 .

sigmaSq

1.086417

0.255371

4.2543 2.097e-05 ***

gamma

0.862306

0.081468 10.5845 < 2.2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -130.5161 cross-sectional data total number of observations = 140 mean efficiency: 0.599406 A likelihood ratio test against the corresponding OLS model indicates that the fit of this SFA model is significantly better than the fit of the corresponding OLS model (without advisory service as explanatory variable): > lrtest( prodCDSfaAdv ) Likelihood ratio test Model 1: OLS (no inefficiency) Model 2: Efficiency Effects Frontier (EEF) #Df

LogLik Df

1

5 -137.61

2

7 -130.52

Chisq Pr(>Chisq)

2 14.185

0.0002907 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

A likelihood ratio test confirms the t-test that the intercept in the inefficiency model is statistically insignificant: > lrtest( prodCDSfaAdv, prodCDSfaAdvInt )

276

6 Stochastic Frontier Analysis Likelihood ratio test Model 1: prodCDSfaAdv Model 2: prodCDSfaAdvInt #Df

LogLik Df Chisq Pr(>Chisq)

1

7 -130.52

2

8 -130.52

1 2e-04

0.9892

The coefficient of the advisory service in the inefficiency model is now significantly negative (at 10% significance level), which means that users of an advisory service have a significantly smaller inefficiency term u, i.e. are significantly more efficient. The size of the coefficients of the inefficiency model (δ) cannot be reasonably interpreted. However, if argument margEff of the efficiencies method is set to TRUE, this method does not only return the efficiency estimates but also the marginal effects of the variables that should explain the efficiency level on the efficiency estimates (see Olsen and Henningsen, 2011): > dat$effCDAdv2 hist( attr( dat$effCDAdv2, "margEff" ), 20 )

0.02

0.03

0.04

0.05

0.06

marginal effect

Figure 6.12: Marginal effects of the variable that indicates the use of an advisory service on the efficiency estimates The resulting graph is shown in figure 6.12. It indicates that apple producers who use an advisory service are between 1.4 and 6.4 percentage points more efficient than apple producers who do not use an advisory service.

6.4 Decomposition of cost efficiency In this section, we estimate a Cobb-Douglas production function and decompose cost efficiency into (cost) technical efficiency and (cost) allocative efficiency. The following code replicates an

277

6 Stochastic Frontier Analysis example in the textbook of Coelli et al. (2005, p. 273, Table 10.2): > # estimating a Copbb-Douglas production frontier (ignoring "other inputs") > riceCdSfaTime summary( riceCdSfaTime, extraPar = TRUE ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 10 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept) 0.2705295

0.0373920

7.2350 4.657e-13 ***

YEARDUM

0.0148901

0.0067910

2.1926

log(area)

0.3557513

0.0597575

5.9533 2.629e-09 ***

log(labor)

0.3507348

0.0635010

5.5233 3.327e-08 ***

log(npk)

0.2565320

0.0352279

7.2821 3.287e-13 ***

sigmaSq

0.2418370

0.0262782

9.2030 < 2.2e-16 ***

gamma

0.8979776

0.0310801 28.8924 < 2.2e-16 ***

sigmaSqU

0.2171642

0.0291291

7.4552 8.970e-14 ***

sigmaSqV

0.0246728

0.0060797

4.0582 4.945e-05 ***

sigma

0.4917693

0.0267180 18.4059 < 2.2e-16 ***

sigmaU

0.4660088

0.0312538 14.9105 < 2.2e-16 ***

sigmaV

0.1570757

0.0193529

8.1164 4.802e-16 ***

lambdaSq

8.8017709

2.9860105

2.9477

lambda

2.9667779

0.5032413

5.8953 3.739e-09 ***

varU

0.0789132

NA

NA

NA

sdU

0.2809149

NA

NA

NA

gammaVar

0.7618134

NA

NA

NA

0.028335 *

0.003202 **

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -83.76704 cross-sectional data total number of observations = 344 mean efficiency: 0.7201092

278

6 Stochastic Frontier Analysis > # extracting coefficients > b0 theta b1 b2 b3 sig2 gamma print( lambda print( sigma # calculating technical efficiencies manually > sig2v sig2u mustar sigstar musig uhati tei # calculating technical efficiencies with the frontier package > tei2 all.equal( tei, tei2 ) [1] TRUE > # calculating cost efficiencies > h2 h3 r ai # cost efficiency due to technical inefficiency > ctei # cost efficiency due to allocative inefficiency > caei = exp( log(r) - ai ) > # cost efficiency > cei # collecting all efficiency measures in a data frame

279

6 Stochastic Frontier Analysis > eff # efficiencies of first and last observations > eff[ c(1:4, 339:344), ] tei

ctei

caei

cei

1

0.7516838 0.7361695 0.9304091 0.6849388

2

0.7404122 0.7245165 0.9494638 0.6879022

3

0.7848997 0.7707317 0.9343774 0.7201543

4

0.8670764 0.8577511 0.8378218 0.7186425

339 0.9362054 0.9321857 0.7034357 0.6557327 340 0.6608281 0.6431254 0.8376981 0.5387449 341 0.7711476 0.7563807 0.9158451 0.6927275 342 0.9260517 0.9212374 0.7758011 0.7146970 343 0.8261157 0.8141105 0.8493577 0.6914710 344 0.8980248 0.8910094 0.8012788 0.7139469 > # some summary statistics of efficiencies > t(sapply(eff, function(x) + +

c( N=length(x), MEAN=mean(x), ST.DEV=sd(x), VARIANCE=var(x), MINIMUM = min(x), MAXIMUM = max(x), COEF.OF.VAR=sd(x)/mean(x)))) N

tei

MEAN

ST.DEV

VARIANCE

MINIMUM

MAXIMUM COEF.OF.VAR

344 0.7201092 0.15320508 0.023471796 0.12701877 0.9574515

0.2127526

ctei 344 0.7059377 0.15662013 0.024529864 0.11600023 0.9550248

0.2218611

caei 344 0.8885442 0.07884834 0.006217061 0.60541374 0.9973617

0.0887388

cei

0.2046942

344 0.6221760 0.12735580 0.016219501 0.08423157 0.8335533

280

7 Data Envelopment Analysis (DEA) 7.1 Preparations We load the R package “Benchmarking” in order to use it for Data Envelopment Analysis: > library( "Benchmarking" ) We create a matrix of input quantities and a vector of output quantities: > xMat yVec deaVrsIn hist( eff( deaVrsIn ) ) Display the “peers” of the first 14 observations: > peers( deaVrsIn )[ 1:14, ] peer1 peer2 peer3 peer4 [1,]

44

73

80

135

[2,]

80

100

126

NA

[3,]

44

54

73

100

[4,]

4

NA

NA

NA

[5,]

17

54

81

NA

[6,]

41

73

126

132

[7,]

7

NA

NA

NA

[8,]

44

54

80

83

[9,]

100

126

132

NA

[10,]

38

73

80

135

[11,]

54

81

100

NA

[12,]

44

54

81

100

[13,]

38

73

80

135

[14,]

44

54

81

100

281

7 Data Envelopment Analysis (DEA) Display the λs of the first 14 observations: > lambda( deaVrsIn )[ 1:14, ] L4 L7

L17 L19

L38

L41

L44

L54 L61 L64

[1,]

0

0 0.00000000

0 0.00000000 0.0000000 0.08707089 0.00000000

0

0

[2,]

0

0 0.00000000

0 0.00000000 0.0000000 0.00000000 0.00000000

0

0

[3,]

0

0 0.00000000

0 0.00000000 0.0000000 0.05466873 0.34157362

0

0

[4,]

1

0 0.00000000

0 0.00000000 0.0000000 0.00000000 0.00000000

0

0

[5,]

0

0 0.07874218

0 0.00000000 0.0000000 0.00000000 0.62716635

0

0

[6,]

0

0 0.00000000

0 0.00000000 0.9520817 0.00000000 0.00000000

0

0

[7,]

0

1 0.00000000

0 0.00000000 0.0000000 0.00000000 0.00000000

0

0

[8,]

0

0 0.00000000

0 0.00000000 0.0000000 0.39228600 0.34818591

0

0

[9,]

0

0 0.00000000

0 0.00000000 0.0000000 0.00000000 0.00000000

0

0

[10,]

0

0 0.00000000

0 0.06541405 0.0000000 0.00000000 0.00000000

0

0

[11,]

0

0 0.00000000

0 0.00000000 0.0000000 0.00000000 0.52820862

0

0

[12,]

0

0 0.00000000

0 0.00000000 0.0000000 0.44076458 0.09749327

0

0

[13,]

0

0 0.00000000

0 0.01725343 0.0000000 0.00000000 0.00000000

0

0

[14,]

0

0 0.00000000

0 0.00000000 0.0000000 0.35937585 0.44329381

0

0

L73 L74 L78

L80

L81

L83

L100 L103

[1,] 0.243735897

0

0 0.6423537 0.00000000 0.00000000 0.0000000

0

[2,] 0.000000000

0

0 0.5147430 0.00000000 0.00000000 0.3620871

0

[3,] 0.153372277

0

0 0.0000000 0.00000000 0.00000000 0.4503854

0

[4,] 0.000000000

0

0 0.0000000 0.00000000 0.00000000 0.0000000

0

[5,] 0.000000000

0

0 0.0000000 0.29409147 0.00000000 0.0000000

0

[6,] 0.002769034

0

0 0.0000000 0.00000000 0.00000000 0.0000000

0

[7,] 0.000000000

0

0 0.0000000 0.00000000 0.00000000 0.0000000

0

[8,] 0.000000000

0

0 0.2101886 0.00000000 0.04933947 0.0000000

0

[9,] 0.000000000

0

0 0.0000000 0.00000000 0.00000000 0.6917918

0

[10,] 0.068686498

0

0 0.2825911 0.00000000 0.00000000 0.0000000

0

[11,] 0.000000000

0

0 0.0000000 0.25455055 0.00000000 0.2172408

0

[12,] 0.000000000

0

0 0.0000000 0.29388540 0.00000000 0.1678567

0

[13,] 0.383969646

0

0 0.5669254 0.00000000 0.00000000 0.0000000

0

[14,] 0.000000000

0

0 0.0000000 0.04033289 0.00000000 0.1569974

0

L126 L129

L132

L135 L137

[1,] 0.000000000

0 0.00000000 0.02683954

0

[2,] 0.123169836

0 0.00000000 0.00000000

0

[3,] 0.000000000

0 0.00000000 0.00000000

0

[4,] 0.000000000

0 0.00000000 0.00000000

0

[5,] 0.000000000

0 0.00000000 0.00000000

0

[6,] 0.008468157

0 0.03668108 0.00000000

0

282

7 Data Envelopment Analysis (DEA) [7,] 0.000000000

0 0.00000000 0.00000000

0

[8,] 0.000000000

0 0.00000000 0.00000000

0

[9,] 0.249102366

0 0.05910586 0.00000000

0

[10,] 0.000000000

0 0.00000000 0.58330837

0

[11,] 0.000000000

0 0.00000000 0.00000000

0

[12,] 0.000000000

0 0.00000000 0.00000000

0

[13,] 0.000000000

0 0.00000000 0.03185153

0

[14,] 0.000000000

0 0.00000000 0.00000000

0

The following commands display the “slack” of the first 14 observations in an input-oriented DEA with VRS: > deaVrsIn table( deaVrsIn$slack ) FALSE

TRUE

78

62

> deaVrsIn$sx[ 1:14, ] sx1 sx2

sx3

[1,]

0

0

0.00000

[2,]

0

0

345.70719

[3,]

0

0

0.00000

[4,]

0

0

0.00000

[5,]

0

0

38.54949

[6,]

0

0

0.00000

[7,]

0

0

0.00000

[8,]

0

0

0.00000

[9,]

0

0

1624.33417

[10,]

0

0

0.00000

[11,]

0

0 12993.07250

[12,]

0

0

0.00000

[13,]

0

0

0.00000

[14,]

0

0

0.00000

> deaVrsIn$sy[ 1:14, ] [1] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 The following command conducts an input-oriented DEA with CRS: > deaCrsIn hist( eff( deaCrsIn ) )

283

7 Data Envelopment Analysis (DEA) We can calculate the scale efficiencies by: > se hist( se ) The following command conducts an input-oriented DEA with DRS > deaDrsIn hist( eff( deaDrsIn ) ) And we check if firms are too small or too large. This is the number of observations that produce at the scale below the optimal scale size: > sum( eff( deaVrsIn ) - eff( deaDrsIn ) > 1e-4 ) [1] 117

7.3 DEA with output-oriented efficiencies The following command conducts an output-oriented DEA with VRS: > deaVrsOut hist( efficiencies( deaVrsOut ) ) The following command conducts an output-oriented DEA with CRS: > deaCrsOut hist( eff( deaCrsOut ) ) In case of CRS, input-oriented efficiencies are equivalent to output-oriented efficiencies: > all.equal( eff( deaCrsIn ), 1 / eff( deaCrsOut ) ) [1] TRUE

7.4 DEA with “super efficiencies” The following command obtains “super efficiencies” for an input-oriented DEA with CRS: > sdeaVrsIn hist( eff( sdeaVrsIn ) )

7.5 DEA with graph hyperbolic efficiencies The following command conducts a DEA with graph hyperbolic efficiencies and VRS: > deaVrsGraph hist( eff( deaVrsGraph ) ) > plot( eff( deaVrsIn ), eff( deaVrsGraph ) ) > abline(0,1)

284

8 Distance Functions 8.1 Theory 8.1.1 Output distance functions The Shepard output distance function is defined as: Do (x, y) = min{λ > 0|y/λ ∈ P (x)}

(8.1)

Do (x, y) = min{λ > 0|(x, y/λ) ∈ T },

(8.2)

or (equivalently) as:

where x is a vector of input quantities, y is a vector of output quantities, P (x) is the production possibility set, and T is the technology set. The Shepard output distance function, Do (x, y), defined in (8.1) and (8.2) returns the Shepard output-oriented technical efficiencies defined in (5.1) and (5.5). Thus, it returns a value of one for fully efficient sets of inputs and outputs (x, y), whereas it returns a non-negative value smaller than one for inefficient sets of inputs and outputs (x, y).1 8.1.1.1 Properties It is usually assumed that the Shepard output distance function Do (x, y) given in (8.1) fulfills the following properties (see, e.g., F¨ are and Primont, 1995; Coelli et al., 2005): 1. Do (x, 0) = 0 for all non-negative x 2. Do (x, y) is non-increasing in x, i.e. ∂Do (x, y)/∂xi ≤ 0 ∀ i 3. Do (x, y) is non-decreasing in y, i.e. ∂Do (x, y)/∂ym ≥ 0 ∀ m 4. Do (x, y) is linearly homogeneous in y, i.e. Do (x, k y) = k Do (x, y) ∀ k > 0 5. Do (x, y) is quasiconvex in x ⇒ convex isoquants2 1

The Farrell output distance function is defined as the inverse of the Shepard output distance function: DoF (x, y) = max{λ > 0|λ y ∈ P (x)} = 1/Do (x, y). It returns the Farrell output-oriented technical efficiencies defined in (5.2) and (5.6). 2 O’Donnell and Coelli (2005) clarified a typographical error in F¨ are and Primont (1995, p. 152) who state by mistake that the output distance function is quasiconcave in input quantities x. The quasiconcavity of the production function is equivalent to the quasiconvexity of the output distance function, whereas the ‘conversion’ between quasiconcavity and quasiconvexity is caused by the opposite ‘direction’ regarding the input quantities

285

8 Distance Functions 6. Do (x, y) is convex in y ⇒ concave transformation curves (i.e., joint production is advantageous compared to specialized production for a given set of input quantities) 7. Do (x, y) ≤ 1 indicates that y belongs to the production possibility set of x, i.e. y ∈ P (x), while Do (x, y) > 1 indicates that y does not belong to the production possibility set of x. 8. Do (x, y) = 1 indicates that y is at the boundary of the production possibility set P (x), i.e. y lies on the transformation curve (given x) and x lies on the isoquant (given y). We will illustrate these properties using a very simple output distance function with two inputs x = (x1 , x2 )0 and two outputs y = (y1 , y2 )0 :3 Do (x, y) =

q

y12 + y22 x−0.4 x2−0.4 1

(8.3)

In the following, we illustrate that this simple output distance function fulfills the abovementioned properties 1–4: ˆ no output

Do (x, 0) =

p

02 + 02 x1−0.4 x2−0.4 = 0

(8.4)

ˆ monotonically non-increasing in x

∂Do (x, y) ∂xi

q

=

−0.4 y12 + y22 x1−0.4 x−0.4 2 xi

≤ 0; i ∈ {1, 2}

(8.5)

ˆ monotonically non-decreasing in y

x−0.4 x−0.4 yi ∂Do (x, y) = 1q 2 ≥ 0; i ∈ {1, 2} ∂yi y12 + y22

(8.6)

ˆ linear homogeneous in y

Do (x, ky) = = =

q

(ky1 )2 + (ky2 )2 x1−0.4 x2−0.4

(8.7)

q

k 2 y12 + k 2 y22 x−0.4 x2−0.4 1

(8.8)

q

k 2 y12 + y22 x1−0.4 x−0.4 2

(8.9)




√ q = k 2 y12 + y22 x−0.4 x2−0.4 1 q

(8.10)

= k y12 + y22 x−0.4 x−0.4 1 2

(8.11)

= kDo (x, y)

(8.12)

(i.e., the production function is increasing in input quantities, while the output distance function is decreasing in input quantities). 3 This output distance function is a very much simplified version of the multiple-output “ray frontier production function” suggested by L¨ othgren (1997, 2000).

286

8 Distance Functions ˆ quasiconvexity in x

A sufficient condition for quasiconvexity is that all leading principal minors of the bordered Hessian matrix are strictly negative (see section 1.5.3). The first derivatives of the output distance function (8.3) with respect to the input quantities are given in (8.5). The remaining elements of the bordered Hessian matrix, i.e. the second derivatives with respect to the input quantities, are: q

∂ 2 Do (x, y)

=

∂xi ∂xj

∆ij =

0.4(∆ij + 0.4) y12 + y22 x1−0.4 x2−0.4 xi xj  1

if i = j

0

if i 6= j

; i ∈ {1, 2}

(8.13)

(8.14)

The first leading principal minor of the bordered Hessian B is: 

|B1 | = −

∂Do (x, y) ∂x1



= − −0.4

q

y12

2

(8.15)

+



= −0.42 y12 + y22

x−1.4 1

y22



x−0.4 2

2

(8.16)

x1−2.8 x−0.8 2

(8.17)

< 0 ∀ x1 , x2 > 0; y1 , y2 ≥ 0; y1 + y2 > 0

(8.18)

The determinant of the (3 × 3) bordered Hessian is: ∂Do (x, y) ∂Do (x, y) ∂ 2 Do (x, y) ∂x1 ∂x2 ∂x1 ∂x2  2 2 o   o ∂Do (x, y) 2 ∂ 2 Do (x, y) ∂D (x, y) ∂ D (x, y) − − ∂x1 ∂x2 ∂x22 ∂x21

|B| =2



= 2 −0.4 

q

y12

+

x−1.4 1

y22

x−0.4 2

q

0.42 y12 + y22 x−1.4 x2−1.4 1 

− −0.4 

− −0.4 

q q

y12 y12

+ +

y22 y22

x−1.4 1 x−0.4 1

x−1.4 2

q

2 −x−1 1



= 0.4

q

y12



−x−1 2

+

y22



x1−0.4

q

y12

x1−0.4

x2−2.4



+

y22

q

y12

x1−2.4

x2−0.4



+

y22

+

2 

0.4 · 1.4 2 

0.4 · 1.4

(8.20)

(8.21) 



−1 0.4 x−1 − −x−1 1 x2 1

x−0.4 2

x−1.4 2

y22

y12

3



x−0.4 1

−0.4

q



x−0.4 2

= 0.4 y12 + y22 x−0.4 x−0.4 1 2  



(8.19)

3 h

2 





1.4 x−2 − −x−1 2 2

2 

−2 −2 −2 −2 −2 0.8 x−2 1 x2 − 1.4 x1 x2 − 1.4 x1 x2

287

1.4 x−2 1 i



(8.22)

8 Distance Functions 

= − 2 0.4

q

y12

= − 2 · 0.43

+

q

y22

x−0.4 1

y12 + y22

3

x−0.4 2

3

−2 x−2 1 x2

(8.23)

x−3.2 x2−3.2 1

(8.24)

< 0 ∀ x1 , x2 > 0; y1 , y2 ≥ 0; y1 + y2 > 0

(8.25)

If both of the input quantities are strictly positive and at least one of the output quantities is strictly positive, the first leading principal minor and the determinant of the bordered Hessian matrix are both strictly negative, which means that both the necessary condition and the sufficient condition for quasiconvexity are fulfilled. Figure 8.1 illustrates the relationship between the two input quantities x1 and x2 and the output distance measure Do (x, y) holding the output quantities y1 and y2 constant so that q √ y12 + y22 = 1, e.g. y1 = y2 = 0.5 (please note that the origin, i.e. x1 = x2 = 0, is in the back). The grey part of the surface indicates distance measures that are larger than one (Do (x, y) > 1), i.e. combinations of the input quantities x1 and x2 that are too small to produce the output quantities y1 and y2 . The dark green line is an isoquant that indicates all combinations of the input quantities x1 and x2 that a fully efficient firm (i.e. Do (x, y) = 1) needs to produce the output quantities y1 and y2 . Given that the output distance function is quasiconvex in x, the lower contour sets, i.e. the sets of input combinations that give an output distance measure (Do (x, y)) smaller than a certain value (e.g. 1), are convex sets (see section 1.5.3). As the origin (x1 = x2 = 0) is not inside the lower contour sets, the quasiconvexity of the output distance function in x results in isoquants that are convex to the origin. The light green part of the surface indicates distance measures (Do (x, y)) that are slightly smaller than one, indicating small technical inefficiency. The red part of the surface indicates distance measures (Do (x, y)) that are considerably smaller than one, indicating considerable technical inefficiency. ˆ convexity in y Convexity requires that the Hessian matrix is positive semidefinite (see section 1.5.2). The elements of the Hessian matrix, i.e. the second derivatives with respect to the output quantities, are: ∂ 2 Do (x, y) x−0.4 x−0.4 yi yj 1 2 ∆ij − 2 = q ; i, j ∈ {1, 2} ∂yi ∂yj y1 + y22 y12 + y22 

∆ij =

 1

if i = j

0

if i 6= j



(8.26)

(8.27)

Equation (8.26) shows that all diagonal elements of the Hessian matrix (i.e. i = j) are non-negative, because 1 − yi2 /(y11 + y22 ) is non-negative for i ∈ {1, 2}.

288

8 Distance Functions

1.0 Do ( x, y )

0.5 0

x2

1

0 1

x1 2

2

Figure 8.1: Output distance function for different input quantities The determinant of the (2 × 2) Hessian matrix is: ∂ 2 Di (x, y) ∂ 2 Di (x, y) |H| = · − ∂x21 ∂x22 x−0.4 2

x−0.4 1 q y12

+

1−

y22

y12

∂ 2 Di (x, y) ∂x1 ∂x2

!

y12 + y22

x−0.4 1 q y12

!2

x2−0.4 +

(8.28)

1−

y22



!

y22 y12 + y22

−

x−0.4 1  q y12

x2−0.4 +

y22

2

y1 y2  y12 + y22

(8.29) 

−0.4

x1 =  q

2

x−0.4 2 

y12 + y22

y2 1− 2 1 2 y1 + y2

!

y2 1− 2 2 2 y1 + y2

!



−0.4

x1 − q

2

x−0.4 2 

y12 + y22



y1 y2 2 y1 + y22

2

(8.30) 

−0.4

x1 =  q

2 "

x−0.4 2 

y12 + y22

2 "



=

x−0.4 x−0.4 2   1 q

y12 + y22

2 "



=

x−0.4 x−0.4 2  1  q

y12 + y22

y2 1− 2 1 2 y1 + y2

!

y2 1− 2 2 2 y1 + y2

!

y22 y12 y12 y22 −
2 y12 + y22 y12 + y22 y12 + y22 y22 y12 y12 + y22


2 −

−

y1 y2 2 y1 + y22

2 #

(8.31)

#

(8.32)

#

y12 y22 y12 + y22




2

=0≥0

(8.33)

As all three principal minors (the two diagonal elements and the determinant) of the Hessian matrix are non-negative, we can conclude that the Hessian matrix is positive semidefinite and, thus, the output distance function 8.3 is convex in y.

289

8 Distance Functions It is indeed not surprising that the determinant of the Hessian matrix is exactly zero, because the output distance function is linearly homogeneous in y and the determinants of linearly homogeneous functions are always zero. Figure 8.2 illustrates the relationship between the two output quantities y1 and y2 and the output distance measure Do (x, y) holding the input quantities x1 and x2 constant so 0.4 that x0.4 1 x2 = 1, e.g. x1 = x2 = 1 (please note that the origin, i.e. y1 = y2 = 0, is in

the front). The grey part of the surface indicates distance measures that are larger than one (Do (x, y) > 1), i.e. combinations of the output quantities y1 and y2 that cannot be produced by the input quantities x1 and x2 . The dark green line is a transformation curve that indicates all combinations of the output quantities y1 and y2 that a fully efficient firm (i.e. Do (x, y) = 1) can produce from the input quantities x1 and x2 . Given that the output distance function is convex in y, it is also quasiconvex in y so that the lower contour sets, i.e. the sets of output combinations that give an output distance measure (Do (x, y)) smaller than a certain value (e.g. 1), are convex sets (see section 1.5.3). As the origin (y1 = y2 = 0) is inside the lower contour sets, the convexity of the output distance function in y results in transformation curves that are concave to the origin. The light green part of the surface indicates distance measures (Do (x, y)) that are slightly smaller than one, indicating small technical inefficiency. The red part of the surface indicates distance measures (Do (x, y)) that are considerably smaller than one, indicating considerable technical inefficiency. 1.5

Do ( x, y )

1.0

0.5

0.0 1.5 1.0

y2

1.5 0.5

1.0 0.5 0.0

y1

0.0

Figure 8.2: Output distance function for different output quantities The following code generates figures 8.1 and 8.2: > nVal colRedGreen x1 x2 dxMat dxFacet dxInt dxCol dxCol[ dxFacet > 1 ] dxMat[ dxMat > dxMat[1,nVal] ] ppx xa lines( trans3d( xa, 1/xa, 1, ppx ), col = "darkgreen", lwd = 5 ) > y1 y2 dyMat dyFacet dyInt dyCol dyCol[ dyFacet > 1 ] dyMat[ dyMat > dyMat[ 1, nVal ] ] ppy ya lines( trans3d( ya, sqrt(1-ya^2), 1, ppy ), col = "darkgreen", lwd = 5 ) 8.1.1.2 Distance elasticities The distance elasticities of the output distance function Do (x, y) with respect to the input quantities (x), I , measure the relative change in the output distance measure (and, hence, in the firm’s efficiency) given a change in one input quantity. The distance elasticity of the ith input is defined as: Ii =

∂Do (x, y) xi ∂ ln Do (x, y) = ≤0 ∂xi Do ∂ ln xi

291

(8.34)

8 Distance Functions The distance elasticities of the output distance function Do (x, y) with respect to output quantities (y), O , measure the relative change in the output distance measure (hence, in the firm’s efficiency) given a change in one output quantity. The distance elasticity of the mth output is defined as: O m =

∂Do (x, y) ym ∂ ln Do (x, y) ≥0 = ∂ym Do ∂ ln ym

(8.35)

Linear homogeneity of the output distance function Do (x, y) in output quantities implies that the distance elasticities with respect to the output quantities sum to one: X

O m =1

(8.36)

m

8.1.1.3 Elasticity of scale The elasticity of scale of an output distance function Do (x, y) equals the negative sum of the distance elasticities with respect to the input quantities: =−

X

Ii

(8.37)

i

8.1.1.4 Optimising behaviour and distance elasticities A necessary condition for revenue maximisation and profit maximisation is that the slope of the transformation curve is equal to the inverse of the negative output price ratio: ∂ym pn =− , ∂yn pm

(8.38)

where pm ; m = 1, . . . , M is the price of the mth output. Equation (8.38) can be re-arranged in the following way:

−

∂ym ∂yn

∂Do (x, y) pn ∂yn = = o ∂D (x, y) pm ∂ym

∂Do (x, y) yn o p n yn ∂yn D (x, y) = o ∂D (x, y) ym pm ym ∂ym Do (x, y) O p n yn n = O p m ym m O n pn yn = O pm ym m

292

(8.39)

(8.40)

(8.41) (8.42)

8 Distance Functions M X

pn yn =

M X O n

pm ym

(8.43)

M pm ym X pn yn = O n O  m n=1 n=1

(8.44)

n=1 M X

M X

O n=1 m

pm ym O m p m ym , = PM n=1 pn yn

pn yn =

(8.45)

O m

(8.46)

n=1

where we use equation (8.36), which is based on the linear homogeneity of the output distance function in output quantities. Hence, in case of revenue maximisation or profit maximisation, the distance elasticities of the outputs are equal to their revenue shares. A necessary condition for cost minimisation and profit maximisation is that the slope of the isoquant is equal to the inverse of the negative input price ratio: ∂xi xj =− , ∂xj xi

(8.47)

where wi ; i = 1, . . . , N is the price of the ith input. Equation (8.47) can be re-arranged in the following way: ∂Do (x, y) ∂xi wj ∂xj − = o (x, y) = ∂D ∂xj wi ∂xi

(8.48)

xj ∂Do (x, y) o wj x j ∂xj D (x, y) = ∂Do (x, y) xi wi x i o ∂xi D (x, y)

(8.49)

Ij Ii

=

wj x j = N X

wj x j =

wj x j wi x i Ij Ii

(8.50)

wi xi

N I X j

(8.51)

wi x i

(8.52)

N wi x i X wj x j = I Ij i j=1 j=1

(8.53)

j=1

I j=1 i

N X

N X

wj x j =

j=1

293

wi x i (−) Ii

(8.54)

8 Distance Functions −Ii wi x i , = PN  j=1 wj xj

(8.55)

where  is the elasticity of scale as derived in equation (8.37). Hence, in case of cost minimisation or profit maximisation, the absolute values of the distance elasticities of the inputs divided by the elasticity of scale are equal to their cost shares. 8.1.1.5 Obtaining unobserved output prices If some (but not all) of the outputs are non-market goods and the assumption of revenue maximisation or profit maximisation is fulfilled, one can use the condition for revenue maximisation or profit maximisation (8.38) to calculate the shadow price of a non-market output by re-arranging equation (8.39) to:

pn = pm

∂Do (x, y) ∂yn , ∂Do (x, y) ∂ym

(8.56)

where pn is the (non-observed) shadow price of a non-market output n and pm is the (observed) price of an output m (see, e.g., Reig-Martinez, Picazo-Tadeo, and Hernandez-Sancho, 2001). PM

If the prices of the individual outputs are unknown but the total revenue (

n=1 pn

yn ) is known

and the assumption of revenue maximisation or profit maximisation is fulfilled, one can use the condition for revenue maximisation or profit maximisation (8.38) to calculate the prices of the individual outputs by re-arranging equation (8.46) to: pm

PM M X O ∂Do (x, y) m n=1 pn yn = pn yn , = ∀m ym n=1 ∂ym Do (x, y)

(8.57)

(see, e.g., F¨ are and Primont, 1995; Reig-Martinez, Picazo-Tadeo, and Hernandez-Sancho, 2001).

8.1.2 Input distance functions The Shepard input distance function is defined as: Di (x, y) = max{λ > 0|x/λ ∈ L(y)}

(8.58)

Di (x, y) = max{λ > 0|(x/λ, y) ∈ T },

(8.59)

or (equivalently) as:

where x is a vector of input quantities, y is a vector of output quantities, L(y) is the input requirement set, and T is the technology set. The Shepard input distance function, Di (x, y), defined in (8.58) and (8.59) returns the Shepard input-oriented technical efficiencies defined in (5.3) and (5.7). Thus, it returns a value of one for fully efficient sets of inputs and outputs (x, y),

294

8 Distance Functions whereas it returns a value larger than one for inefficient sets of inputs and outputs (x, y).4 8.1.2.1 Properties It is usually assumed that the Shepard input distance function Di (x, y) defined in (8.58) fulfills the following properties (see, e.g., F¨ are and Primont, 1995; Coelli et al., 2005): 1. Di (x, y) is non-decreasing in x, i.e. ∂Di (x, y)/∂xi ≥ 0 ∀ i 2. Di (x, y) is non-increasing in y, i.e. ∂Di (x, y)/∂ym ≤ 0 ∀ m 3. Di (x, y) is linearly homogeneous in x, i.e. Di (k x, y) = k Di (x, y) ∀ k > 0 4. Di (x, y) is concave in x ⇒ convex isoquants 5. Di (x, y) is quasiconcave in y ⇒ concave transformation curves5 (i.e., joint production is advantageous compared to specialized production for a given set of input quantities) 6. Di (x, y) ≥ 1 indicates that x belongs to the input requirement set of y, i.e. x ∈ L(y), while Di (x, y) < 1 indicates that x does not belong to the input requirement set of y. 7. Di (x, y) = 1 indicates that x is at the boundary of the input requirement set L(y), i.e. x lies on the isoquant (given y) and y lies on the transformation curve (given x). We will illustrate these properties using a very simple input distance function with two inputs x = (x1 , x2 )0 and two outputs y = (y1 , y2 )0 :6 x0.5 x0.5 Di (x, y) = q1 2 y12 + y22

(8.60)

In the following, we illustrate that this simple input distance function fulfills the abovementioned properties 1–3: ˆ monotonically non-decreasing in x 0.5 x0.5 ∂Di (x, y) 1 x2 = 0.5 q ≥ 0; i ∈ {1, 2} ∂xi x y2 + y2 i

1

(8.61)

2

ˆ monotonically non-increasing in y

x0.5 x0.5 yi ∂Di (x, y) = − 1 2
1.5 ≤ 0; i ∈ {1, 2} ∂yi y12 + y22 4

(8.62)

The Farrell input distance function is defined as the inverse of the Shepard input distance function: DiF (x, y) = min{λ > 0|λ x ∈ L(y)} = 1/DiF (x, y). It returns the Farrell input-oriented technical efficiencies defined in (5.4) and (5.8). 5 O’Donnell and Coelli (2005) clarified a typographical error in F¨ are and Primont (1995, p. 152) who state by mistake that the input distance function is quasiconvex in output quantities y. 6 The specification of this input distance function is inspired by the multiple-output “ray frontier production function” suggested by L¨ othgren (1997, 2000).

295

8 Distance Functions ˆ linear homogeneous in x

Di (kx, y) =

=

(k x1 )0.5 (k x2 )0.5 q

(8.63)

y12 + y22

0.5 x0.5 k 0.5 x0.5 1 k 2

(8.64)

q

y12 + y22

k x0.5 x0.5 = q1 2 y12 + y22

(8.65)

= k Di (x, y)

(8.66)

ˆ concave x

Concavity requires that the Hessian matrix is negative semidefinite (see section 1.5.2). The elements of the Hessian matrix, i.e. the second derivatives with respect to the input quantities, are: 0.5 ∂ 2 Di (x, y) x0.5 1 x2 q = 0.5 (0.5 − ∆ij ) ∂xi ∂xj xi xj y12 + y22

∆ij =

 1

if i = j

0

if i 6= j

(8.67)

(8.68)

Equation (8.67) shows that all diagonal elements of the Hessian matrix (i.e. i = j) are non-positive, because (0.5 − ∆ii ) is negative for i ∈ {1, 2}, while all other terms on the right-hand side of (8.67) are always non-negative. The determinant of the (2 × 2) Hessian matrix is: ∂ 2 Di (x, y) ∂ 2 Di (x, y) |H| = · − ∂x21 ∂x22 = 0.5 (−0.5)

0.5 x0.5 1 x2

q

∂ 2 Di (x, y) ∂x1 ∂x2

· 0.5 (−0.5)

x21 y12 + y22

!2

x10.5 x0.5 2 q

x22 y12 + y22

(8.69) 

− 0.5 · 0.5

x10.5 x20.5 q

x1 x2 y12 + y22

2 

(8.70) =

0.54 x1 x2 y12 + y22


−

0.54 x1 x2 y12 + y22


=0≥0

(8.71)

As all first-order principal minors (i.e. the two diagonal elements) are non-positive and the second-order principal (i.e. the determinant) is non-negative, we can conclude that the Hessian matrix is negative semidefinite and, thus, the input distance function (8.60) is concave in x. It is indeed not surprising that the determinant of the Hessian matrix is exactly zero, because the input distance function is linearly homogeneous in x and the determinants of linearly homogeneous functions are always zero.

296

8 Distance Functions Figure 8.3 illustrates the relationship between the two input quantities x1 and x2 and the input distance measure Di (x, y) holding the output quantities y1 and y2 constant so that q √ y12 + y22 = 1, e.g. y1 = y2 = 0.5 (please note that the origin, i.e. x1 = x2 = 0, is in the front). The grey part of the surface indicates distance measures that are smaller than one (Di (x, y) < 1), i.e. combinations of the input quantities x1 and x2 that are too small to produce the output quantities y1 and y2 . The dark green line is an isoquant that indicates all combinations of the input quantities x1 and x2 that a fully efficient firm (i.e. Di (x, y) = 1) needs to produce the output quantities y1 and y2 . Given that the input distance function is concave in x, it is also quasiconcave in x so that the upper contour sets, i.e. the sets of input combinations that give an input distance measure (Di (x, y)) larger than a certain value (e.g. 1), are convex sets (see section 1.5.3). As the origin (x1 = x2 = 0) is not inside the upper contour sets (only on its border if Di (x, y) = 0), the concavity of the input distance function in x results in isoquants that are convex to the origin. The light green part of the surface indicates distance measures (Di (x, y)) that are slightly larger than one, indicating small technical inefficiency. The red part of the surface indicates distance measures (Di (x, y)) that are considerably larger than one, indicating considerable technical inefficiency.

2 Di ( x, y )

1

0 4

x2

2

4 2

x1 0

0

Figure 8.3: Input distance function for different input quantities ˆ quasiconcavity in y

A sufficient condition for quasiconcavity of a function with two arguments is that the first leading principal minor of the bordered Hessian matrix is strictly negative, while its second leading principle minor is strictly positive (see section 1.5.3). The first derivatives of the input distance function (8.60) with respect to the output quantities are given in (8.62). The remaining elements of the bordered Hessian matrix, i.e. the second derivatives with respect

297

8 Distance Functions to the output quantities, are: x10.5 x20.5 ∂ 2 Di (x, y) =
1.5 ∂yi ∂yj y12 + y22 ∆ij =



 1

if i = j

0

if i 6= j

3 yi yj − ∆ij ; i ∈ {1, 2} y12 + y22 

(8.72)

(8.73)

The first leading principal minor of the bordered Hessian B is: ∂Do (x, y) |B1 | = − ∂x1 

=− − =−

2

(8.74)

0.5 x0.5 1 x2 y1

y12 + y22 x1 x2 y12

y12 + y22

!2

(8.75)


1.5

(8.76)


3

< 0 ∀ x1 , x2 > 0; y1 > 0; y2 ≥ 0

(8.77)

The determinant of the (3 × 3) bordered Hessian is: |B| =2

∂Di (x, y) ∂Di (x, y) ∂ 2 Di (x, y) ∂y1 ∂y2 ∂y1 ∂y2

−

∂Di (x, y) ∂y1

=2 − − − − − = = =

!2

∂ 2 Di (x, y) − ∂y22 !

0.5 x0.5 1 x 2 y1

y12 + y22

0.5 x0.5 1 x2 y1

y12 + y22

y12 + y22


2 4.5

y12 + y2 1.5 x1.5 1 x2 y12 + y22

y12 + y22 !2

6 y12 y22 − y12 y12 + y22

h


1.5


1.5

0.5 x0.5 1 x2

!3 "

!

!

0.5 x0.5 1 x2

y12 + y22


1.5

1.5 x1.5 1 x2

y12 + y22


1.5

y12 + y22

∂Di (x, y) ∂y2

x10.5 x20.5 y2

!2


1.5

0.5 x0.5 1 x2 y2

0.5 x0.5 1 x2

−


1.5

(8.78)

y12 + y22


1.5

!2

∂ 2 Di (x, y) ∂y12

3 x10.5 x20.5 y1 y2 y12 + y22

!!

3 y12 −1 y12 + y22

!!

3 y22 − 1 − y22 y12 + y22

i

(8.79)


2.5

3 y22 −1 y12 + y22

!

!

!#

3 y12 −1 y12 + y22

(8.80) (8.81) (8.82)


3.5

> 0 ∀ x1 , x2 > 0; y1 , y2 ≥ 0; y1 + y2 > 0

(8.83)

If both of the input quantities are strictly positive and at least one of the output quantities is strictly positive. the first leading principal minor of the bordered Hessian matrix is non-

298

8 Distance Functions positive and its determinant is non-negative, which means that the necessary conditions for quasiconvexity are fulfilled. If the quantities of both of the inputs and of the first output are strictly positive and the quantity of the second output is non-negative, the first leading principal minor of the bordered Hessian matrix is strictly negative and its determinant is strictly positive, which means that also the sufficient conditions for quasiconvexity are fulfilled. As the order of the outputs is arbitrary, we can rearrange the order of the outputs so that the quantity of the first input is strictly positive as long as at least one output quantity is strictly positive. Hence, if at least one of the output quantities is strictly positive, not only the necessary conditions but also the sufficient conditions for quasiconvexity are fulfilled. Figure 8.4 illustrates the relationship between the two output quantities y1 and y2 and the input distance measure Di (x, y) holding the input quantities x1 and x2 constant so 0.5 that x0.5 1 x2 = 1, e.g. x1 = x2 = 1 (please note that the origin, i.e. y1 = y2 = 0, is in

the back). The grey part of the surface indicates distance measures that are smaller than one (Di (x, y) < 1), i.e. combinations of the output quantities y1 and y2 that cannot be produced by the input quantities x1 and x2 . The dark green line is a transformation curve that indicates all combinations of the output quantities y1 and y2 that a fully efficient firm (i.e. Di (x, y) = 1) can produce from the input quantities x1 and x2 . Given that the input distance function is quasiconcave in y, the upper contour sets, i.e. the sets of output combinations that give an input distance measure (Di (x, y)) larger than a certain value (e.g. 1), are convex sets (see section 1.5.3). As the origin (y1 = y2 = 0) is inside the upper contour sets, the quasiconcavity of the input distance function in y results in transformation curves that are concave to the origin. The light green part of the surface indicates distance measures (Di (x, y)) that are slightly larger than one, indicating small technical inefficiency. The red part of the surface indicates distance measures (Di (x, y)) that are considerably larger than one, indicating considerable technical inefficiency. The following code generates figures 8.3 and 8.4: > nVal colRedGreen x1 x2 dxMat dxFacet dxInt dxCol dxCol[ dxFacet < 1 ] dxMat[ dxMat > 3 ] ppx xa lines( trans3d( xa, 1/xa, 1, ppx ), col = "darkgreen", lwd = 5 ) > y1 y2 dyMat dyFacet dyInt dyCol dyCol[ dyFacet < 1 ] # dyMat[ dyMat > dyMat[ 1, nVal ] ] dyMat[ dyMat > 2 ] ppy ya lines( trans3d( ya, sqrt(1-ya^2), 1, ppy ), col = "darkgreen", lwd = 5 )

300

8 Distance Functions 8.1.2.2 Distance elasticities The distance elasticities of the input distance function Di (x, y) with respect to input quantities (x), I , measure the relative change in the input distance measure (and, hence, in the firm’s inefficiency) given a change in one input quantity. The distance elasticity of the ith input quantity is defined as: Ii =

∂Di (x, y) xi ∂ ln Di (x, y) = ≥0 ∂xi Di ∂ ln xi

(8.84)

Linear homogeneity of the input distance function Di (x, y) in input quantities implies that its distance elasticities with respect to the input quantities sum to one: X

Ii = 1

(8.85)

i

The distance elasticities of the input distance function Di (x, y) with respect to output quantities (y), O , measure the relative change in the input distance measure (and, hence, in the firm’s inefficiency) given a change in one output quantity. The distance elasticity of the mth output quantity is defined as: O m =

∂Di (x, y) ym ∂ ln Di (x, y) = ≤0 ∂ym Di ∂ ln ym

(8.86)

8.1.2.3 Elasticity of scale The elasticity of scale of an input distance function Di (x, y) equals the inverse of the negative sum of distance elasticities with respect to the output quantities: !−1

=

−

X

O m

(8.87)

m

8.2 Cobb-Douglas output distance function 8.2.1 Specification The general form of the Cobb-Douglas output distance function is Do (x, y) = A

N Y

!

xαi i

i=1

M Y

! βm ym ,

(8.88)

m=1

which can be linearized to: ln Do (x, y) = α0 +

N X

αi ln xi +

i=1

with α0 = ln A.

301

M X m=1

βm ln ym

(8.89)

8 Distance Functions

8.2.2 Estimation The Cobb-Douglas output distance function can be expressed (and then estimated) as traditional stochastic frontier model. In the following, we will derive, under which conditions the Cobb-Douglas output distance function defined in (8.88) and (8.89) is linear homogeneous in output quantities (y): kDo (x, y) = Do (x, ky)

(8.90)

ln(kDo (x, y)) = ln Do (x, ky) ln(kDo (x, y)) = α0 + ln k + ln Do (x, y) = α0 +

N X i=1 N X

(8.91)

αi ln xi + αi ln xi +

i=1

ln k + ln Do (x, y) = ln Do (x, y) + ln k

M X m=1 M X

βm ln(k ym ) βm ln k +

m=1 M X

M X

(8.92) βm ln ym

(8.93)

m=1

βm

(8.94)

m=1

ln k = ln k

M X

βm

(8.95)

m=1

1=

M X

βm

(8.96)

m=1

Hence, the Cobb-Douglas output distance function is linear homogeneous in output quantities if the coefficients of the output quantities (βm ) sum up to one. We can impose linear homogeneity in output quantities by re-arranging equation (8.96) to: βM = 1 −

M −1 X

βm

(8.97)

m=1

and by substituting the right-hand side of (8.97) for βM in (8.89): o

ln D (x, y) = α0 + ln Do (x, y) = α0 + − ln yM = α0 +

N X i=1 N X i=1 N X i=1

αi ln xi + αi ln xi + αi ln xi +

M −1 X m=1 M −1 X m=1 M −1 X

βm ln(ym ) + 1 −

M −1 X

!

βm

ln(yM )

(8.98)

m=1

βm (ln ym − ln yM ) + ln yM βm ln(ym /yM ) − ln Do (x, y).

(8.99) (8.100)

m=1

We can assume that u ≡ − ln(Do (x, y)) ≥ 0 follows a half normal or truncated normal distribution (i.e. u ∼ N + (µ, σu2 )) and add a disturbance term v, that accounts for statistical noise and

302

8 Distance Functions follows a normal distribution (i.e. v ∼ N (0, σv2 )) so that we get: − ln(yM ) = α0 +

N X

αi ln xi +

i=1

M −1 X

βm ln(ym /yM ) + u + v.

(8.101)

m=1

This specification is equivalent to the specification of stochastic frontier models so that we can use the stochastic frontier methods introduced in Chapter 6 to estimate this output distance function. In the following, we will estimate a Cobb-Douglas output distance function with our data set of French apple producers. This data set distinguishes between two outputs: the quantity of apples produced (qApples) and the quantity of the other outputs (qOtherOut). The following command estimates the Cobb-Douglas output distance function using the command sfa() for stochastic frontier analysis: > odfCD summary( odfCD ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency increases the endogenous variable (as in a cost function) The dependent variable is logged Iterative ML estimation terminated after 11 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept)

10.776633

1.393814

7.7318 1.061e-14 ***

log(qCap)

-0.087286

0.086890 -1.0046

0.315106

log(qLab)

-0.501954

0.157395 -3.1891

0.001427 **

log(qMat)

-0.451892

0.116505 -3.8787

0.000105 ***

log(qOtherOut/qApples)

0.621158

0.044476 13.9663 < 2.2e-16 ***

sigmaSq

1.306200

0.219877

gamma

0.923605

0.039930 23.1307 < 2.2e-16 ***

5.9406 2.840e-09 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -147.8225 cross-sectional data total number of observations = 140

303

8 Distance Functions mean efficiency: 0.5069105

8.2.3 Distance elasticities The distance elasticities of a Cobb-Douglas output distance function with respect to input quantities (x), I , can be calculated as: Ii =

∂Do (x, y) xi ∂ ln Do (x, y) = = αi ∂xi Do ∂ ln xi

(8.102)

The distance elasticities of a Cobb-Douglas output distance function with respect to output quantities (y), O , can be calculated as: O m =

∂Do (x, y) ym ∂ ln Do (x, y) = βm = ∂ym Do ∂ ln ym

(8.103)

Since the distance elasticities of Cobb-Douglas output distance functions are equal to some of the estimated parameters, they can be directly obtained from the estimation output. The distance elasticity with respect to the capital input (IqCap = −0.087) indicates that a 1% increase in the capital input results (ceteris paribus) in a 0.09% decrease of the distance measure (i.e. efficiency decreases and inefficiency increases). The distance elasticity with respect to the labor input (IqLab = −0.502) indicates that a 1% increase in the labor input results (ceteris paribus) in a 0.5% decrease of the distance measure (i.e. efficiency decreases and inefficiency increases). The distance elasticity with respect to the material input (IqMat = −0.452) indicates that a 1% increase in the materials input results (ceteris paribus) in a 0.45% decrease of the distance measure (i.e. efficiency decreases and inefficiency increases). The distance elasticity with respect to the other output (O qOtherOut = 0.621) indicates that a 1% increase in other outputs results (ceteris paribus) in a 0.62% increase of the distance measure (i.e. efficiency increases and inefficiency decreases). Based on the homogeneity property (8.97), O we can calculate the distance elasticity of the apple quantity O qApples = 1 − qOtherOut = 0.379,

which indicates that a 1% increase in the output of apples results (ceteris paribus) in a 0.38% increase of the distance measure (i.e. efficiency increases and inefficiency decreases).

8.2.4 Elasticity of scale The elasticity of scale for an output distance function Do (x, y) equals the negative sum of distance elasticities with respect to the input quantities: =−

X

Ii

(8.104)

i

For our Cobb-Douglas output distance function, the elasticity of scale can be obtained by using following command: > -sum(coef(odfCD)[c("log(qCap)", "log(qLab)", "log(qMat)")])

304

8 Distance Functions [1] 1.041133 The estimated elasticity of scale  = 1.04 indicates slightly increasing returns to scale.

8.2.5 Properties 8.2.5.1 Non-increasing in input quantities Do (x, y) is non-increasing in x if its partial derivatives with respect to the input quantities are non-positive: Do (x, y) ∂ ln Do (x, y) Do (x, y) ∂Do (x, y) = = αi ≤0 ∂xi ∂ ln xi xi xi

(8.105)

Given that Do (x, y) and xi are always non-negative, a Cobb-Douglas output distance function Do (x, y) is non-increasing in x if all coefficients αi are non-positive. We can see from the summary() output of our estimated Cobb-Douglas output distance function that all coefficients related to input quantities are negative. This means that monotonicity in input quantities is fulfilled. 8.2.5.2 Non-decreasing in output quantities Do (x, y) is non-decreasing in y if its partial derivatives with respect to the output quantities are non-negative: ∂Do (x, y) ∂ ln Do (x, y) Do (x, y) Do (x, y) = = βm ≥0 ∂ym ∂ ln ym ym ym

(8.106)

Given that Do (x, y) and ym are always non-negative, a Cobb-Douglas output distance function Do (x, y) is non-decreasing in y if all coefficients βm are non-negative. For our Cobb-Douglas output distance function, the coefficient of the (normalized) quantity of “other outputs” is positive (0.621). The coefficient of the apple quantity can be recovered from the homogeneity condition (1-0.621 = 0.379). Since the coefficients of both output quantities are positive, we can conclude that monotonicity in output quantities is fulfilled. 8.2.5.3 Quasiconvex in input quantities In the following, we check if our estimated Cobb-Douglas output distance function is quasiconvex in input quantities (x). A sufficient condition for quasiconvexity of Do (x, y) in x is that |Bi | < 0; i = 1, . . . , N , where |Bi | is the ith leading principal minor of the bordered Hessian matrix B

305

8 Distance Functions with





0

f1

f2

...

fN

  f1   B=  f2  .  .. 

f11

f12

...

f12 .. .

f22 .. .

... .. .

f1N  

fN

f1N

f2N

. . . fN N



f2N .. .

 ,    

(8.107)

where fi = ∂Do (x, y)/∂xi are the first-order partial derivatives and fij are the second-order partial derivatives of the Cobb-Douglas output distance function Do (x, y) with respect to the input quantities x: 

o





o

∂ ∂D∂x(x,y) ∂ αi D x(x,y) ∂ 2 Do (x, y) i i fij = = = ∂xi ∂xj ∂xj ∂xj αi ∂Do (x, y) Do (x, y) = − δij αi xi ∂xi x2i o Do (x, y) αi D (x, y) − δij αi = αj xi xj x2i Do (x, y) = αi (αj − δij ) , xi xj



(8.108) (8.109) (8.110) (8.111)

where δij is (again) Kronecker’s delta (2.95). As all elements of the bordered Hessian matrix (both the first derivatives and the second derivatives) contain the output distance measure Do (x, y) as a multiplicative element, we can factor it out of the bordered Hessian matrix. As the output distance measure Do (x, y) is nonnegative, it does not affect the signs of the principal minors so that we can ignore it in our further calculations. For a sake of simplicity of the code, we create short-cuts for the coefficients. > oCap oLab oMat dat$fCap dat$fLab dat$fMat dat$fCapCap dat$fLabLab dat$fMatMat dat$fCapLab dat$fCapMat dat$fLabMat bhmArray bhmArray[ 1, 2, ] det( hessianArray[ , , 1 ] ) [1] -4.801944e-18 As the two diagonal elements (first-order principal minors) of the Hessian matrix are strictly negative, we can conclude that the Hessian matrix is not positive semidefinite and, thus, our estimated Cobb-Douglas output distance function is not convex in output quantities at the first observation. It is not surprising that the determinant of the Hessian matrix with respect to the output quantities is exactly zero or very close to zero, because the output distance function is linearly homogeneous in output quantities, which implies that the determinant of the Hessian must be zero. The following code checks the semidefiniteness of the Hessian matrix at each individual observation:

309

8 Distance Functions > dat$convexOutOdfCD table( dat$convexOutOdfCD ) FALSE 140 This shows that our estimated Cobb-Douglas output distance function is convex in output quantities y not at a single observation. Indeed, Cobb-Douglas output distance functions cannot be convex in output quantities, because this functional form implies transformation curves that are convex to the origin.

8.2.6 Efficiency estimates If there were no inefficiencies, the output distance measures Do (x, y) would be equal to one (which would imply that u = − ln Do = − ln 1 = 0) and all observations would be on the frontier. We can use a likelihood ratio test to investigate if there is significant inefficiency. > lrtest( odfCD ) Likelihood ratio test Model 1: OLS (no inefficiency) Model 2: Error Components Frontier (ECF) #Df

LogLik Df

1

6 -156.41

2

7 -147.82

Chisq Pr(>Chisq)

1 17.179

1.701e-05 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The very small p-value confirms that there is significant inefficiency. The efficiency estimates can be obtained by the efficiencies method: > dat$effOdfCD summary( dat$effOdfCD ) efficiency Min.

:0.02638

1st Qu.:0.33639 Median :0.54408 Mean

:0.50691

3rd Qu.:0.67876 Max.

:0.89680

310

8 Distance Functions A median Shepard output-oriented efficiency estimate of 0.544 indicates that this apple producer produces 45.6% less of each output quantity than would be possible to produce with its input quantities.

8.3 Translog output distance function 8.3.1 Specification The general form of the Translog output distance function is: ln Do (x, y) = α0 +

N X

αi ln xi +

i=1 M X

+ +

βm ln ym +

m=1 N X M X

N X N 1X αij ln xi ln xj 2 i=1 j=1

(8.117)

M X M 1 X βmn ln ym ln yn 2 m=1 n=1

ζim ln xi ln ym

i=1 m=1

with αij = αji ∀ i, j and βmn = βnm ∀ n, m.

8.3.2 Estimation The Translog output distance function can be expressed (and then estimated) as traditional stochastic frontier model. This is facilitated by imposing linear homogeneity in the output quantities (y). In the following, we will derive, under which conditions the Translog output distance function defined in (8.117) is linear homogeneous in output quantities (y): k Do (x, y) =Do (x, k y)

(8.118)

ln(kDo (x, y)) = ln Do (x, ky)

(8.119)

ln(kDo (x, y)) =α0 +

N X

αi ln xi +

i=1

+ +

M X

βm ln(k ym ) +

m=1 N X M X

ln k + ln Do (x, y) =α0 +

αi ln xi +

i=1 M X m=1

(8.120)

M X M 1 X βmn ln(k ym ) ln(k yn ) 2 m=1 n=1

ζim ln xi ln(k ym )

i=1 m=1 N X

+

N X N 1X αij ln xi ln xj 2 i=1 j=1

βm ln(k) +

N X N 1X αij ln xi ln xj 2 i=1 j=1 M X m=1

311

βm ln ym

(8.121)

8 Distance Functions

+

M X M M X M 1 X 1 X βmn ln(k) ln(k) + βmn ln(k) ln(yn ) 2 m=1 n=1 2 m=1 n=1

M X M M X M 1 X 1 X + βmn ln(ym ) ln(k) + βmn ln(ym ) ln(yn ) 2 m=1 n=1 2 m=1 n=1

+

N X M X

ζim ln xi ln k +

i=1 m=1

N X M X

ζim ln xi ln ym

i=1 m=1

M X M 1 X βmn ln k + ln D (x, y) = ln D (x, y) + ln(k) βm + ln(k) ln(k) 2 m=1 n=1 m=1 o

M X

o

(8.122)

M M M M X X 1X 1 X + ln(k) ln(yn ) βmn + ln(k) ln(ym ) βmn 2 n=1 2 m=1 m=1 n=1

+ ln k

N X

i=1 M X

ln k = ln(k)

ln xi

M X

ζim

m=1

βm + ln(k) ln(k)

m=1

+ ln(k) + ln k M X

N X

ln xi

+

M X

ζim m=1 M X M X

βm + ln(k)

m=1

+

(8.123)

M M M M X X 1 X 1X ln(yn ) βmn + ln(k) ln(ym ) βmn 2 n=1 2 m=1 m=1 n=1

i=1

1=

M X M 1 X βmn 2 m=1 n=1

1 βmn 2 m=1 n=1

(8.124)

M M M M X X 1 X 1X βmn + βmn ln(yn ) ln(ym ) 2 n=1 2 m=1 m=1 n=1 N X

ln xi

i=1

M X

ζim

m=1

This condition is only fulfilled for all values of k, ym , and xi , if the parameters fulfill the following conditions: M X m=1 M X m=1 M X

βm = 1

(8.125) βmn =βnm

βmn = 0 ∀ n ←−−−−−−→

M X

βmn = 0 ∀ m

(8.126)

n=1

ζim = 0 ∀ i

(8.127)

m=1

In order to impose linear homogeneity in output quantities, we can rearrange these restrictions

312

8 Distance Functions to get: βM = 1 − βM n = − βmN = − ζiM = −

M −1 X

βm

m=1 M −1 X m=1 M −1 X

(8.128)

βmn ∀ n

(8.129)

βmn ∀ m

(8.130)

n=1 M −1 X

ζim ∀ i

(8.131)

m=1

By substituting the right-hand sides of equations (8.128) to (8.131) for βM , βM n , βmN , and ζiM in equation (8.117) and re-arranging this equation, we get: − ln yM =α0 +

N X

αi ln xi +

i=1

+

M −1 X

βm ln

m=1

+

−1 N M X X

N X N 1X αij ln xi ln xj 2 i=1 j=1

(8.132)

−1 −1 M X ym 1 MX yn ym βmn ln + ln yM 2 m=1 n=1 yM yM

ζim ln xi ln

i=1 m=1

ym − ln Do (x, y). yM

We can assume that u ≡ − ln(Do (x, y)) ≥ 0 follows a half-normal or truncated normal distribution (i.e. u ∼ N + (µ, σu2 ) ) and add a disturbance term v that accounts for statistical noise and follows a normal distribution (i.e. v ∼ N (0, σv2 ) ) so that we get: − ln yM =α0 +

N X

αi ln xi +

i=1

+ +

M −1 X

βm m=1 −1 N M X X

N N X 1X αij ln xi ln xj 2 i=1 j=1

(8.133)

−1 M −1 X ym 1 MX ym yn ln + βmn ln ln yM 2 m=1 n=1 yM yM

ζim ln xi ln

i=1 m=1

ym + u + v. yM

(8.134)

This specification is equivalent to the specification of stochastic frontier models so that we can use the stochastic frontier methods introduced in Chapter 6 to estimate this output distance function. In the following, we use command sfa() to estimate a Translog output distance function for our data set of French apple producers: > odfTL summary( odfTL ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency increases the endogenous variable (as in a cost function) The dependent variable is logged Iterative ML estimation terminated after 28 iterations: cannot find a parameter vector that results in a log-likelihood value larger than the log-likelihood value obtained in the previous step final maximum likelihood estimates Estimate Std. Error z value (Intercept)

Pr(>|z|)

38.882431

21.847091

1.7798 0.0751164

log(qCap)

0.901531

2.138098

0.4217 0.6732797

log(qLab)

-5.046845

4.402014 -1.1465 0.2515943

log(qMat)

-1.509307

3.738472 -0.4037 0.6864165

I(0.5 * log(qCap)^2)

0.068372

0.186448

0.3667 0.7138388

I(0.5 * log(qLab)^2)

1.185172

0.654824

1.8099 0.0703096

I(0.5 * log(qMat)^2)

0.204052

0.368668

0.5535 0.5799313

I(log(qCap) * log(qLab))

-0.475477

I(log(qCap) * log(qMat))

0.401302

I(log(qLab) * log(qMat))

-0.458295

0.409950 -1.1179 0.2635971

log(qOtherOut/qApples)

-0.076972

0.784813 -0.0981 0.9218712

I(0.5 * log(qOtherOut/qApples)^2)

0.130865

0.275217 -1.7276 0.0840518 0.194155

0.018159

2.0669 0.0387422

7.2065 5.740e-13

I(log(qCap) * log(qOtherOut/qApples)) -0.022233

0.052472 -0.4237 0.6717711

I(log(qLab) * log(qOtherOut/qApples))

0.028802

0.081368

0.3540 0.7233620

I(log(qMat) * log(qOtherOut/qApples))

0.057849

0.071400

0.8102 0.4178179

sigmaSq

0.677662

0.175928

3.8519 0.0001172

gamma

0.820876

0.135616

6.0530 1.422e-09

(Intercept)

.

log(qCap) log(qLab) log(qMat) I(0.5 * log(qCap)^2)

314

8 Distance Functions I(0.5 * log(qLab)^2)

.

I(0.5 * log(qMat)^2) I(log(qCap) * log(qLab))

.

I(log(qCap) * log(qMat))

*

I(log(qLab) * log(qMat)) log(qOtherOut/qApples) I(0.5 * log(qOtherOut/qApples)^2)

***

I(log(qCap) * log(qOtherOut/qApples)) I(log(qLab) * log(qOtherOut/qApples)) I(log(qMat) * log(qOtherOut/qApples)) sigmaSq

***

gamma

***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -116.8208 cross-sectional data total number of observations = 140 mean efficiency: 0.6032496 We can use a likelihood ratio test to compare the Translog output distance function with the corresponding Cobb-Douglas output distance function: > lrtest( odfCD, odfTL ) Likelihood ratio test Model 1: odfCD Model 2: odfTL #Df 1 2

LogLik Df

Chisq Pr(>Chisq)

7 -147.82 17 -116.82 10 62.003

1.511e-09 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The likelihood ratio test clearly rejects the Cobb-Douglas output distance function in favor of the Translog output distance function.

315

8 Distance Functions

8.3.3 Distance elasticities The distance elasticities of a Translog output distance function with respect to input quantities (x), I , can be obtained by: Ii =

N M X X ∂Do (x, y) xi ∂ ln Do (x, y) = α + α ln x + ζim ln ym = i ij j ∂xi Do ∂ ln xi m=1 j=1

(8.135)

The distance elasticities of a Translog output distance function with respect to output quantities (y), O , can be obtained by: O m =

M N X X ∂Do (x, y) ym ∂ ln Do (x, y) = β + β ln y + ζim ln xi = m mn n ∂ym Do ∂ ln ym n=1 i=1

(8.136)

In order to facilitate the calculation of the distance elasticities, we create short-cuts for the estimated coefficients: > aCap aLab aMat aCapCap aCapLab abline( 1, 0 ) > plot( dat$X, dat$eScaleOdfTL, log = "x" )

1.6 1.4 1.2 0.8

1.0

dat$eScaleOdfTL

1.4 1.2 1.0 0.8

dat$eScaleOdfTL

1.6

> abline( 1, 0 )

1e+05

5e+05

2e+06

1e+07

0.5

dat$qOut

1.0

2.0

5.0

dat$X

Figure 8.5: Estimated elasticities of scale for different farm sizes The resulting graphs are shown in figure 8.5. According to these estimates of the elasticity of scale, farms with an aggregate output quantity of less than 1,000,000 units and with an aggregate input quantity of less than the sample mean generally have increasing returns to scale so that increasing their size would increase their (total factor) productivity. In contrast, farms with an aggregate output quantity of more than 5,000,000 units and with an aggregate input quantity of more than 1.5 times the sample mean generally have decreasing returns to scale so that increasing their size would decrease their (total factor) productivity.

318

8 Distance Functions

8.3.5 Properties 8.3.5.1 Non-increasing in input quantities In order to investigate the monotonicity of Translog output distance function in input quantities x, we calculate the partial derivatives (fi ; i = 1, . . . , N ) of the Translog output distance function with respect to the input quantities (xi ): fi =

∂Do (x, y) ∂ ln Do (x, y) Do (x, y) Do (x, y) = = Ii ∀i. ∂xi ∂ ln xi xi xi

(8.137)

As the output distance measure Do (x, y) and the input quantities xi are always non-negative, we can check the monotonicity in input quantities by checking the sign of the distance elasticities with respect to the input quantities: > table( dat$eCapOdfTL table( dat$eLabOdfTL table( dat$eMatOdfTL dat$monoInOdfTL table( dat$eOtherOdfTL >= 0 ) FALSE

TRUE

2

138

> dat$monoOutOdfTL = 0 & dat$eOtherOdfTL >= 0 > table( dat$monoOutOdfTL ) FALSE

TRUE

9

131

Combining the monotonicity conditions for inputs and outputs, we can investigate how many observations fulfill all monotonicity conditions: > dat$monoOdfTL table( dat$monoOdfTL ) FALSE

TRUE

63

77

8.3.5.3 Quasiconvex in input quantities In the following, we will check if our estimated Translog output distance function is quasiconvex in input quantities. A sufficient condition for quasiconvexity of Do (x, y) in x is that |Bi | < 0; i = 1, . . . , N , where |Bi | is the ith leading principal minor of the bordered Hessian matrix B with 



0

f1

f2

...

fN

  f1   B=  f2  .  .. 

f11

f12

...

f12 .. .

f22 .. .

... .. .

f1N  

fN

f1N

f2N

. . . fN N



f2N .. .

 ,    

(8.139)

where fi = ∂Do (x, y)/∂xi are the first-order partial derivatives and fij are the second-order partial derivatives of the Translog output distance function Do (x, y) with respect to the input quantities x. We use equation (8.137) and the following R code to calculate the (first-order) partial derivatives of the Translog output distance function with respect to the input quantities x at the “frontier” (i.e. Do (x, y) = 1) for each observation:

320

8 Distance Functions > dat$fCapOdfTL dat$fLabOdfTL dat$fMatOdfTL summary( dat[ , c( "fCapOdfTL", "fLabOdfTL", "fMatOdfTL" ) ] ) fCapOdfTL Min.

:-8.168e-05

fLabOdfTL Min.

fMatOdfTL

:-1.941e-05

Min.

:-2.370e-04

1st Qu.:-4.015e-06

1st Qu.:-4.461e-06

1st Qu.:-4.396e-05

Median :-1.759e-06

Median :-2.165e-06

Median :-2.649e-05

Mean

Mean

Mean

:-4.040e-06

:-3.099e-06

:-3.557e-05

3rd Qu.: 2.070e-07

3rd Qu.:-5.804e-07

3rd Qu.:-1.203e-05

Max.

Max.

Max.

: 8.224e-06

: 4.679e-06

: 7.724e-06

We can calculate the second-order derivatives with respect to input quantities with following formula: o

D ∂( ∂ ln∂Dln x(x,y) ∂ 2 Do (x, y) ∂fi i fij = = = ∂xi ∂xj ∂xj ∂xj

∂ =



αi +

o (x,y)

PN

xi

j=1 αij ln xj +

)

(8.140)

PM

m=1 ζim ln ym



Do (x,y) xi

∂xj o o αi j D (x, y) ∂ ln D (x, y) ∂ ln Do (x, y) Do (x, y) = + xi xi ∂ ln xi ∂ ln xj xi xj o o ∂ ln D (x, y) D (x, y) − δij ∂ ln xj xi xj Do (x, y) Do (x, y) = αij + Ii Ij xi xj xi xj o D (x, y) − δij Ij xi xj   D o (x, y) = αij + Ii Ij − δij Ij xi xj



(8.141) (8.142) (8.143) (8.144) (8.145) (8.146)

We use the following R code to compute the second-order derivatives of the Translog output distance function with respect to input quantities x for each observation: > dat$fCapCapOdfTL dat$fLabLabOdfTL dat$fMatMatOdfTL dat$fCapLabOdfTL det( hessianArray[ , , 1 ] ) [1] 6.718547e-18 As the two diagonal elements (first-order principal minors) of the Hessian matrix are strictly negative, we can conclude that the Hessian matrix is not positive semidefinite and, thus, our estimated Translog output distance function is not convex in output quantities at the first observation. It is not surprising that the determinant of the Hessian matrix with respect to the output quantities is exactly zero or very close to zero, because the output distance function is linearly homogeneous in output quantities, which implies that the determinant of the Hessian must be zero. The following code checks the semidefiniteness of the Hessian matrix at each individual observation: > dat$convexOutOdfTL table( dat$convexOutOdfTL )

325

8 Distance Functions FALSE

TRUE

119

21

> table( dat$convexOutOdfTL[ dat$monoOdfTL ] ) FALSE

TRUE

71

6

This shows that our estimated Translog output distance function is not convex in output quantities y at 119 out of 140 observations.

8.3.6 Efficiency estimates If there were no inefficiencies, the output distance measure Do of all observations would be equal to 1 (which implies u = − ln Do = 0) and all observations would be on the frontier. We can use a likelihood ratio test to investigate if there is inefficiency: > lrtest( odfTL ) Likelihood ratio test Model 1: OLS (no inefficiency) Model 2: Error Components Frontier (ECF) #Df

LogLik Df

1

16 -117.87

2

17 -116.82

Chisq Pr(>Chisq)

1 2.1065

0.07334 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The efficiency estimates can be obtained by the efficiencies() method: > dat$effOdfTL summary( dat$effOdfTL ) efficiency Min.

:0.1494

1st Qu.:0.4773 Median :0.6359 Mean

:0.6032

3rd Qu.:0.7348 Max.

:0.8843

The following command creates a histogram of the efficiency estimates: > hist( dat$effOdfTL, 20 ) The resulting graph is shown in figure 8.6.

326

10 0

5

Frequency

15

20

8 Distance Functions

0.2

0.4

0.6

0.8

dat$effOdfTL

Figure 8.6: Efficiency estimates of the Translog output distance function

8.4 Cobb-Douglas input distance function 8.4.1 Specification The general form of the Cobb-Douglas input distance function is: N Y

i

D (x, y) = A

!

xαi i

i=1

M Y

! βm ym

,

(8.155)

βm ln ym

(8.156)

m=1

which can be linearized to: i

ln D (x, y) = α0 +

N X

αi ln xi +

i=1

M X m=1

with α0 = ln A.

8.4.2 Estimation The Cobb-Douglas input distance function can be expressed (and then estimated) as traditional stochastic frontier model. In the following, we will derive, under which conditions the Cobb-Douglas input distance function defined in (8.155) and (8.156) is linear homogeneous in input quantities (x): kDi (x, y) = Di (kx, y)

(8.157)

ln(kDi (x, y)) = ln Di (kx, y)

327

(8.158)

8 Distance Functions

ln(kDi (x, y)) = α0 + i

ln k + ln D (x, y) = α0 +

N X

M X

αi ln k ln xi +

i=1 M X

βm ln ym

(8.159)

m=1

αi ln k +

m=1

ln k + ln Di (x, y) = ln Di (x, y) + ln k

N X

M X

αi ln xi +

βm ln ym

(8.160)

m=1

i=1 N X

αi

(8.161)

i=1

ln k = ln k

N X

αi

(8.162)

i=1

1=

N X

αi

(8.163)

i=1

Hence, the Cobb-Douglas input distance function is linear homogeneous in input quantities if the coefficients of the input quantities (αi ) sum up to one. We can impose linear homogeneity in input quantities by re-arranging equation (8.163) to: αN = 1 −

N −1 X

αi

(8.164)

i=1

and by substituting the right-hand side of (8.164) for αN in (8.156): ln Di (x, y) = α0 + ln Di (x, y) = α0 + ln Di (x, y) − ln xN = α0 + − ln xN = α0 +

N −1 X i=1 N −1 X i=1 N −1 X i=1 N −1 X

αi ln xi + (1 −

N −1 X

αi ) ln xN +

M X

βm ln(ym )

(8.165)

m=1

i=1

αi (ln xi − ln xN ) + ln xi +

M X

βm ln yM

(8.166)

m=1

αi (ln xi /xN ) + αi (ln xi /xN ) +

M X m=1 M X

βm ln ym

(8.167)

βm ln ym − ln Di (x, y)

(8.168)

m=1

i=1

We can assume that u ≡ ln(Di (x, y)) ≥ 0 follows a half normal or truncated normal distribution (i.e. u ∼ N + (µ, σu2 )) and add a disturbance term v that accounts for statistical noise and follows a normal distribution (i.e. v ∼ N (0, σv2 )) so that we get: − ln(xN ) = α0 +

N −1 X

αi ln(xi /xN ) +

M X

βm ln ym − u + v

(8.169)

m=1

i=1

This specification is equivalent to the specification of stochastic frontier models so that we can use the stochastic frontier methods introduced in Chapter 6 to estimate this input distance function. In the following, we will estimate a Cobb-Douglas input distance function with our data set of French apple producers. This data set distinguishes between two outputs: the quantity of

328

8 Distance Functions apples produced (qApples) and the quantity of the other outputs (qOtherOut). The following command estimates the Cobb-Douglas input distance function using the command sfa() for stochastic frontier analysis: > idfCD summary( idfCD ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 10 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error (Intercept)

-11.084030

z value

Pr(>|z|)

0.135311 -81.9155 < 2.2e-16 ***

log(qCap/qMat)

0.029982

0.040619

0.7381

0.4604

log(qLab/qMat)

0.684137

0.065195

10.4938 < 2.2e-16 ***

log(qApples)

-0.158068

0.022508

-7.0229 2.173e-12 ***

log(qOtherOut)

-0.129456

0.028820

-4.4918 7.061e-06 ***

sigmaSq

0.375209

0.065546

5.7243 1.038e-08 ***

gamma

0.925826

0.044667

20.7271 < 2.2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -60.46615 cross-sectional data total number of observations = 140 mean efficiency: 0.6594516

8.4.3 Distance elasticities The distance elasticities of a Cobb-Douglas input distance function Di (x, y) with respect to input quantities (x), I , can be calculated as: Ii =

∂Di (x, y) xi ∂ ln Di (x, y) = = αi i ∂xi D ∂ ln xi

329

(8.170)

8 Distance Functions The distance elasticities of a Cobb-Douglas input distance function Di (x, y) with respect to output quantities (y), O , can be calculated as: O m =

∂Di (x, y) ym ∂ ln Di (x, y) = βm = ∂ym Di ∂ ln ym

(8.171)

Since the distance elasticities of Cobb-Douglas input distance functions are equal to some of the estimated parameters, they can be directly obtained from the estimation output. The distance elasticity with respect to the capital input (IqCap = 0.03) indicates that a 1% increases in the capital input results (ceteris paribus) in a 0.03% increase of the distance measure (i.e., inefficiency increases and efficiency decreases). The distance elasticity with respect to the labor input (IqLab = 0.684) indicates that a 1% increases in the labor input results (ceteris paribus) in a 0.68% increase of the distance measure (i.e., inefficiency increases and efficiency decreases). The distance elasticity with respect to the material input (IqMat = 0.286) indicates that a 1% increase in the material input results (ceteris paribus) in a 0.29% increase of the distance measure (i.e., inefficiency increases and efficiency decreases). The distance elasticity with respect to the apples output (O qApples = −0.158 indicates that a 1% increase in the apple output results (ceteris paribus) in a 0.16% decrease of the distance measure (i.e., inefficiency decreases and efficiency increases). The distance elasticity with respect to the other outputs (O qOther = −0.129 indicates that a 1% increase in the other outputs results (ceteris paribus) in a 0.13% decrease of the distance measure (i.e., inefficiency decreases and efficiency increases).

8.4.4 Elasticity of scale The elasticity of scale for an input distance function Di (x, y) equals the inverse of the negative sum of the distance elasticities with respect to the output quantities: !−1

=

−

X

O m

(8.172)

m

For our Cobb-Douglas input distance function, the elasticity of scale can be obtained by using following command: > (-sum(coef(idfCD)[c("log(qApples)", "log(qOtherOut)")]))^(-1) [1] 3.477971 The estimated elasticity of scale  = 3.478 indicates highly increasing returns to scale.

330

8 Distance Functions

8.4.5 Properties 8.4.5.1 Non-decreasing in input quantities Di (x, y) is non-decreasing in x if its partial derivatives with respect to the input quantities are non-negative: Di (x, y) ∂ ln Di (x, y) Di (x, y) ∂Di (x, y) = = αi ≥ 0. ∂xi ∂ ln xi xi xi

(8.173)

Given that Di (x, y) and xi are always non-negative, a Cobb-Douglas input distance function fulfills the monotonicity condition, if all coefficients of the (normalized) input quantities are positive. This is the case for our estimated Cobb-Douglas input distance function, where the coefficient of the quantity of the material input (qMat) that was not directly estimated can be obtained from the homogeneity condition, i.e., 1 − 0.03 − 0.684 = 0.286. 8.4.5.2 Non-increasing in output quantities Di (x, y) is non-increasing in y if its partial derivatives with respect to the output quantities are non-positive: ∂Di (x, y) ∂ ln Di (x, y) Di (x, y) Di (x, y) = = βm ≤ 0. ∂ym ∂ ln ym ym ym

(8.174)

Given that Di (x, y) and xi are always non-negative, a Cobb-Douglas input distance function is non-increasing in output quantities if the coefficients of all output quantities are non-positive. This is the case for our estimated Cobb-Douglas input distance function so that we can conclude that it is non-increasing in output quantities. 8.4.5.3 Concave in input quantities In the following, we check if the estimated Cobb-Douglas input distance function is concave in input quantities (x). Di (x, y) is concave in x if and only if its Hessian matrix H with respect to the input quantities is negative semidefinite. A sufficient condition for negative semidefiniteness of a symmetric matrix is that all its ith-order principal minors (not only its leading principal minors) are non-positive for i being odd and non-negative for i being even for all i ∈ {1, . . . , N } (see section 1.5.2). The Hessian matrix with respect to the input quantities is 

f11

f12

...

f1N



  f12  H= .  .. 

f22 .. .

... .. .

f2N .. .

   ,  

f1N

f2N

. . . fN N

331

(8.175)

8 Distance Functions where fij are the second derivatives of Di (x, y) with respect to input quantities x: i

D ∂( ∂ ln∂Dln x(x,y) ∂ 2 Di (x, y) ∂fi i fij = = = ∂xi ∂xj ∂xj ∂xj

i (x,y)

xi

)

(8.176)

=

αi ∂Di (x, y) Di (x, y) − δij αi xi ∂xi x2i

(8.177)

=

Di (x, y) αi Do (x, y) αj − δij αi xi xj x2i

(8.178)

= αi (αj − δij )

Di (x, y) xi xj

(8.179)

where δij is (again) Kronecker’s delta (2.95). As all elements of the Hessian matrix (all second derivatives) contain the input distance measure Di (x, y)

as a multiplicative element, we can factor it out of the Hessian matrix. As the input

distance measure Di (x, y) is non-negative, it does not affect the signs of the principal minors so that we can ignore it in our further calculations. Again, for the sake of simplicity of the code, we create short-cuts for the coefficients: > iCap iLab iMat dat$fCapCap dat$fLabLab dat$fMatMat dat$fCapLab dat$fCapMat dat$fLabMat hessianArray hessianArray[ 1, 1, ] hessianArray[ 1, 2, ] det( hessianArray[ -2, -2, 1 ] ) [1] 1.029466e-22 > det( hessianArray[ -1, -1, 1 ] ) [1] 5.609556e-24 > det( hessianArray[ , , 1 ] ) [1] -1.574506e-50 As all first-order principal minors (diagonal elements) are negative, all second-order principal minors are positive, and the determinant (third-order principal minor) is virtually zero, the conditions for concavity are fulfilled at the first observation. The following code checks the concavity at all observations: > dat$CDidfConcaveX table( dat$CDidfConcaveX ) TRUE 140 The estimated Cobb-Douglas input distance function is concave in input quantities at all observations.

333

8 Distance Functions 8.4.5.4 Quasiconcave in output quantities In the following, we check if the estimated Cobb-Douglas input distance function Di (x, y) is quasiconcave in output quantities (y). A sufficient condition for quasiconcavity in y is that |B1 | < 0, |B2 | > 0, |B3 | < 0, . . . , (−1)N |BN | > 0, where |Bi | is the ith principal minor of the bordered Hessian matrix B with 



0

f1

f2

...

fN

  f1   B=  f2  .  .. 

f11

f12

...

f12 .. .

f22 .. .

... .. .

f1N  

fN

f1N

f2N

. . . fN N



f2N .. .

 ,    

(8.180)

where fm = ∂Di (x, y)/∂ym are the first partial derivatives and fmn are the second partial derivatives of the input distance function with respect to the output quantities y: 

fmn

i





i

∂ ∂D∂y(x,y) ∂ βm D y(x,y) ∂ 2 Di (x, y) m m = = = ∂ym ∂yn ∂yn ∂yn βm ∂Di (x, y) Di (x, y) = − δij βm 2 ym ∂yn ym βm Di (x, y) Di (x, y) = βn − δij βm 2 ym yn ym i D (x, y) = βm (βn − δij ) ym yn



(8.181) (8.182) (8.183) (8.184)

where δij is (again) Kronecker’s delta (2.95). As all elements of the bordered Hessian matrix (both the first derivatives and the second derivatives) contain the input distance measure Di (x, y) as a multiplicative element, we can factor it out of the bordered Hessian matrix. As the output distance measure Di (x, y) is nonnegative, it does not affect the signs of the principal minors so that we can ignore it in our further calculations. In order to facilitate further computations, we create short-cuts for the coefficients regarding the output quantities y: > iApples iOther dat$fApples dat$fOther dat$fApplesApples dat$fApplesOther dat$fOtherOther bhmArray bhmArray[ 1, 2, ] dat$CDidfQuasiConcaveY 0

> table( dat$CDidfQuasiConcaveY ) FALSE 140 This estimated Cobb-Douglas input distance function is quasiconcave in output quantities (y) not at a single observation in our data set. Indeed, Cobb-Douglas input distance functions cannot be quasiconcave in output quantities, because this functional form implies transformation curves that are convex to the origin.

335

8 Distance Functions

8.4.6 Efficiency estimates If there were no inefficiencies the input distance Di would be equal to 1 (what would imply that ln Di (1) = 0, i.e., u = 0) and all observations would be on the frontier. We can use likelihood ratio test to investigate if there is inefficiency > lrtest( idfCD ) Likelihood ratio test Model 1: OLS (no inefficiency) Model 2: Error Components Frontier (ECF) #Df

LogLik Df

1

6 -66.739

2

7 -60.466

Chisq Pr(>Chisq)

1 12.546

0.0001986 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The small p-value indicates that there is statistically significant inefficiency. The efficiency estimates can be obtained by the efficiencies method. By default, the efficiencies method calculates the efficiency estimates as Td E = E[e−u ]. If we set argument minusU of the efficiencies method to FALSE, it calculates the efficiency estimates as Td E = E[eu ].7 Hence, in order to obtain Shepard input-oriented efficiency estimates from an input distance function, we need to set argument minusU of the efficiencies method to FALSE: > dat$effIdfCD summary( dat$effIdfCD ) efficiency Min.

:1.069

1st Qu.:1.244 Median :1.493 Mean

:1.718

3rd Qu.:1.887 Max.

:5.207

A median Shepard input-oriented efficiency estimate of 1.493 indicates that this apple producer uses 49.3% more of each input quantity than would be necessary to produce its output quantities.

7

Please note that E[eu ] is not equal to E[e−u ]


−1

.

336

8 Distance Functions

8.5 Translog input distance function 8.5.1 Specification The general form of the Translog input distance function is: ln Di (x, y) =α0 +

N X

αi ln xi +

i=1

+ +

(8.185)

M X M 1 X βmn ln ym ln yn ln ym + 2 m=1 n=1

M X

βm m=1 N X M X

N X N 1X αij ln xi ln xj 2 i=1 j=1

ζim ln xi ln ym ,

i=1 m=1

with αij = αji ∀ i, j and βmn = βnm ∀ n, m.

8.5.2 Estimation The Translog input distance function can be expressed (and then estimated) as traditional stochastic frontier model. This is facilitated by imposing linear homogeneity in the input quantities (x). In the following, we will derive, under which conditions the Translog input distance function defined in (8.185) is linear homogeneous in input quantities (x): k Di (x, y) =Di (k x, y)

(8.186)

ln(kDi (x, y)) = ln Di (k x, y)

(8.187)

ln(kDi (x, y)) =α0 +

N X

αi ln(k xi ) +

i=1

+ +

M X

i=1 m=1 N X

i

ln k + ln D (x, y) =α0 + 1 + 2 +

(8.188)

M M X 1 X βmn ln ym ln yn 2 m=1 n=1

βm ln yi +

m=1 N X M X

N X N 1X αij ln(k xi ) ln(k xj ) 2 i=1 j=1

ζim ln(k xi ) ln ym

αi ln k i=1 N X N X

+

N X

αi ln xi

(8.189)

i=1

N X M 1X αij ln k ln k + αij ln k ln xj 2 i=1 j=1 i=1 j=1

N X N N X N 1X 1X αij ln xi ln k + αij ln xi ln xj 2 i=1 j=1 2 i=1 j=1

337

(8.190)

8 Distance Functions M X

+ +

βm ln ym +

m=1 N X M X

M X M 1 X αmn ln ym ln yn 2 m=1 n=1

ζim ln k ln ym +

i=1 m=1

N X M X

ζim ln xi ln ym

i=1 m=1

N X N 1X ln k + ln D (x, y) = ln D (x, y) + ln k αi + ln k ln k αij 2 i=1 j=1 i=1 i

N X

i

+ ln k

N N N N X X 1X 1X ln xj αij + ln k ln xi αij 2 j=1 2 i=1 i=1 j=1 M X

+ ln k

ln ym

m=1

N X

ζim

i=1

N X

N X N 1X ln k = ln k αi + ln k ln k αij 2 i=1 j=1 i=1

+ ln k

M X

ln ym

m=1

1=

(8.192)

N N N M X X 1X 1X ln xj αij + ln k ln xi αij 2 j=1 2 i=1 i=1 j=1

+ ln k N X

(8.191)

αi + ln k

i=1

N X

ζim

i=1 N N X 1X

2

αij

(8.193)

i=1 j=1

N N N N X X 1X 1X αij + αij ln xj ln xi 2 j=1 2 i=1 i=1 j=1

+

M X

+

ln ym

m=1

N X

ζim

i=1

This condition is only fulfilled for all values of k, xi , and ym if the parameters fulfill the following conditions: N X i=1 N X

αi = 1

(8.194) αij =αji

αij = 0 ∀ j ←−−−−→

i=1 N X

N X

αij = 0 ∀ i

(8.195)

j=1

ζim = 0 ∀ m

(8.196)

i=1

In order to impose linear homogeneity in input quantities, we can rearrange these restrictions to get: αN = 1 −

N −1 X i=1

338

αi

(8.197)

8 Distance Functions N −1 X

αN j = − αiN = −

i=1 N −1 X

αij ∀ j

(8.198)

αij ∀ i

(8.199)

j=1

ζN m = −

N −1 X

ζim ∀ m

(8.200)

i=1

By substituting the right-hand sides of equations (8.197) to (8.200) for αN , αN j , αiN , and ζN m in equation (8.185) and re-arranging this equation, we get: − ln xN =α0 +

N −1 X

αi ln

i=1

+ +

M X

−1 N −1 X 1 NX xj xi xi + ln αij ln xN 2 i=1 j=1 xN xN

βm ln ym +

m=1 N −1 X M X

ζim ln

i=1 m=1

(8.201)

M X M 1 X βmn ln ym ln yn 2 m=1 n=1

xi ln ym − ln Di (x, y). xN

We can assume that u ≡ ln(Di (x, y)) ≥ 0 follows a half-normal or truncated normal distribution (i.e. u ∼ N + (µ, σu2 ) ) and add a disturbance term v that accounts for statistical noise and follows a normal distribution (i.e. v ∼ N (0, σv2 ) ) so that we get: − ln xN =α0 +

N −1 X

αi ln

i=1

+ +

M X

−1 −1 N X xi xi 1 NX xj αij ln + ln xN 2 i=1 j=1 xN xN

βm ln ym +

m=1 M N −1 X X

ζim ln

i=1 m=1

(8.202)

M X M 1 X βmn ln ym ln yn 2 m=1 n=1

xi ln ym − u + v. xN

This specification is equivalent to the specification of stochastic frontier models so that we can use the stochastic frontier methods introduced in Chapter 6 to estimate this input distance function. In the following, we use command sfa() to estimate a Translog output distance function for our data set of French apple producers: > idfTL summary( idfTL ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 22 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate (Intercept)

-10.6399124

Std. Error

z value

Pr(>|z|)

0.3968280 -26.8124 < 2.2e-16

log(qCap/qMat)

-0.4212393

0.2066687

-2.0382

0.04153

log(qLab/qMat)

0.5768280

0.4168776

1.3837

0.16645

I(0.5 * log(qCap/qMat)^2)

0.1343118

0.0840011

1.5989

0.10984

I(0.5 * log(qLab/qMat)^2)

-0.0586589

0.2218504

-0.2644

0.79147

0.1436734

0.1014410

1.4163

0.15668

log(qApples)

-0.3003764

0.0709196

-4.2355 2.281e-05

log(qOtherOut)

-0.2566521

0.1475028

-1.7400

I(0.5 * log(qApples)^2)

-0.0842968

0.0110531

-7.6266 2.411e-14

I(0.5 * log(qOtherOut)^2)

-0.0204688

0.0269612

-0.7592

0.44774

I(log(qApples) * log(qOtherOut))

0.0127001

0.0181271

0.7006

0.48354

I(log(qCap/qMat) * log(qApples))

0.0535355

0.0296150

1.8077

0.07065

I(log(qCap/qMat) * log(qOtherOut))

0.0081670

0.0461674

0.1769

0.85959

I(log(qLab/qMat) * log(qApples))

0.0038709

0.0267088

0.1449

0.88477

I(log(qLab/qMat) * log(qOtherOut))

0.0406196

0.0701908

0.5787

0.56279

sigmaSq

0.2280426

0.0438838

5.1965 2.031e-07

gamma

0.9048670

0.0599052

15.1050 < 2.2e-16

I(log(qCap/qMat) * log(qLab/qMat))

(Intercept)

***

log(qCap/qMat)

*

log(qLab/qMat) I(0.5 * log(qCap/qMat)^2) I(0.5 * log(qLab/qMat)^2) I(log(qCap/qMat) * log(qLab/qMat)) log(qApples)

***

340

0.08186

8 Distance Functions log(qOtherOut)

.

I(0.5 * log(qApples)^2)

***

I(0.5 * log(qOtherOut)^2) I(log(qApples) * log(qOtherOut)) I(log(qCap/qMat) * log(qApples))

.

I(log(qCap/qMat) * log(qOtherOut)) I(log(qLab/qMat) * log(qApples)) I(log(qLab/qMat) * log(qOtherOut)) sigmaSq

***

gamma

***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -29.39859 cross-sectional data total number of observations = 140 mean efficiency: 0.7180246 We can use a likelihood ratio test to compare the Translog input distance function with the corresponding Cobb-Douglas input distance function: > lrtest( idfCD, idfTL ) Likelihood ratio test Model 1: idfCD Model 2: idfTL #Df 1 2

LogLik Df

Chisq Pr(>Chisq)

7 -60.466 17 -29.399 10 62.135

1.426e-09 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The likelihood ratio test clearly rejects the Cobb-Douglas input distance function in favor of the Translog input distance function.

341

8 Distance Functions

8.5.3 Distance elasticities The distance elasticities of a Translog input distance function with respect to the input quantities (x), I , can be calculated as: Ii =

N M X X ∂Di (x, y) xi ∂ ln Di (x, y) = α + α ln x + ζim ln ym = i ij j ∂xi Di ∂ ln xi m=1 j=1

(8.203)

The distance elasticities of a Translog input distance function with respect to the output quantities (y), O , can be calculated as: O m =

M N X X ∂Di (x, y) ym ∂ ln Di (x, y) = β + β ln y + ζim ln xi = m mn n ∂ym Di ∂ ln ym n=1 i=1

(8.204)

In order to facilitate the calculation of the distance elasticities, we create short-cuts for the estimated coefficients: > aCap aLab aMat aCapCap aCapLab Chisq)

1 5.1635

0.01153 *

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The test rejects the hypothesis that all firms are fully technically efficient at 5% significance level. The Shepard input-oriented efficiency estimates can be obtained by the efficiencies method: > dat$effIdfTL summary( dat$effIdfTL ) efficiency Min.

:1.059

1st Qu.:1.190 Median :1.348 Mean

:1.498

3rd Qu.:1.719 Max.

:2.937

8.5.6 Imposing monotonicity The monotonicity conditions can be written as: N M X X ∂ ln Di (x, y) = αi + αij ln xj + ζim ln ym ≥ 0 ∀ i = 1, . . . , N ∂ ln xi m=1 j=1

344

(8.205)

8 Distance Functions M N X X ∂ ln Di (x, y) = βm + βmn ln yn + ζim ln xi ≤ 0 ∀ m = 1, . . . , M ∂ ln ym n=1 i=1

(8.206)

Replacing the non-estimated coefficients by the right-hand sides of equations (8.197) to (8.200) and multiplying the monotonicity restrictions regarding the outputs by −1, we get: N −1 M X X ∂ ln Di (x, y) = αi + αij ln xj + αiN ln xN + ζim ln ym ∂ ln xi m=1 j=1

= αi +

N −1 X



αij ln xj + −

N −1 X

j=1

= αi +

N −1 X



j=1

ζim ln ym

M X xj + ζim ln ym ≥ 0 ∀ i = 1, . . . , N xN m=1

(8.209)

N −1 M X X ∂ ln Di (x, y) xj = αN + αN j ln + ζN m ln ym ∂ ln xN xN m=1 j=1

=1−

N −1 X

αi +

i=1

=1−

αi −

αi −

i=1

−

!

αij

i=1

−1 N −1 N X X

αij ln

i=1 j=1

i=1

⇔−

−

N −1 X

j=1

N −1 X

N −1 X

N −1 X

N −1 N −1 X X i=1 j=1

(8.210)

M N −1 X X xj ln + − ζim ln ym xN m=1 i=1

!

M N −1 X X xj ζim ln ym ≥ 0 − xN i=1 m=1

N −1 X M X xj αij ln − ζim ln ym ≥ −1 xN i=1 m=1

N −1 M X X ∂ ln Di (x, y) βmn ln yn − ζim ln xi − ζN m ln xN = −βm − ∂ ln ym n=1 i=1

= −βm − = −βm −

M X n=1 M X n=1

βmn ln yn − βmn ln yn −

N −1 X i=1 N −1 X

(8.208)

m=1

j=1

αij ln

M X

αij  ln xN +

(8.207)

ζim ln xi − −

N −1 X

(8.211)

(8.212)

(8.213)

(8.214) !

ζim ln xN

(8.215)

i=1

ζim ln

i=1

xi ≥0 xN

(8.216)

We can write these monotonicity restrictions using matrix notation as: R θ ≥ r,

(8.217)

where R = [Rα , Rβ , Rζ ] with

(8.218)

345

8 Distance Functions

Rα =



0

1

0

...

0

ln xxN1

ln xxN2

...

                    

0 .. .

0 .. .

1 .. .

... .. .

0 .. .

0 .. .

ln xxN1 .. .

... .. .

0

0

0

...

1

0

0 −1 −1 . . . −1 − ln

− ln

x1 xN

x2 xN

+ ln



xN −1 xN

0 .. . ln xxN1

...

0 

x1 xN

ln



. . . − ln

x1 xN

+ ln

xN −1 xN



0

...

0

...

0



ln xxN2 .. .

... .. .

ln xxn−1 N .. .

... .. .

0 .. .

0

...

...

xN −1 xN x − xNN−1

                    

− ln

x2 xN

... −



x2 xN x2 xN

+

xN −1 xN



...

0

0

0

...

0

0

0

...

0

0

...

0

...

0

0 .. .

0 .. .

0 .. .

... .. .

0 .. .

0 .. .

0 .. .

... .. .

0 .. .

0 .. .

... .. .

0 .. .

... .. .

0 .. .

0

0

0

...

0

0

0

...

0

0

...

0

...

0

(8.219)

Rβ =



0

0

...

0

0

0

...

0

0

...

0

...

0



                    

0 .. .

0 .. .

... .. .

0 .. .

0 .. .

0 .. .

... .. .

0 .. .

0 .. .

... .. .

0 .. .

... .. .

0 .. .

0

0

...

0

0

0

...

0

0

...

0

...

0

0

0

...

0

0

0

...

0

0

...

0

...

0

−1

0

...

0

0

...

0

...

0

−1 . . . .. . . . .

0 .. .

... .. .

0 .. .

                    



Rζ =

0 .. . 0

0

− ln y1 . . . .. .. . .

0 .. .

. . . −1

ln y

1   0   ..   .   0    − ln y1    − ln x1 xN    0   ..  . 

− ln y1 − ln y2 . . . − ln yM

0

0

...

0 .. . − ln y1

0

...

−lny2

. . . − ln yM

ln y2

...

ln yM

0

0

...

0

...

0

0

...

0



0 .. .

... .. .

0 .. .

ln y1 .. .

ln y2 .. .

... .. .

ln yM .. .

... .. .

0 .. .

0 .. .

... .. .

0 .. .

0

...

0

...

0

                    

...

ln y1

ln y2

...

ln yM

. . . − ln yM

...

− ln y1

− ln y2

...

− ln yM

0

...

0

. . . − ln

0

...

0

0 .. .

− ln xxN2 .. .

... .. .

0 .. .

... .. .

0 .. .

.. .

... .. .

0 .. .

0

0

...

0

0

. . . − ln

0

0

. . . − ln yM

− ln y1

− ln y2

0

...

0

− ln xxN2

− ln xxN1 .. .

... .. .

0 .. .

− ln y2

0

− ln y2 . . . −lnyM .. .. .. . . .

(8.220)

. . . − ln xxN1

0

. . . − ln xxN2

xN −1 xN

− ln

xN −1 xN

xN −1 xN

(8.221)

is the restriction matrix for the monotonicity restrictions (∂ ln Di (x, y)/∂ ln x1 , ∂ ln Di (x, y)/∂ ln x2 , . . . , ∂ ln Di (x, y)/∂ ln xN −1 , ∂ ln Di (x, y)/∂ ln xN , ∂ ln Di (x, y)/∂ ln y1 , ∂ ln Di (x, y)/∂ ln y2 , . . . , ∂ ln Di (x, y)/∂ ln yM ), θ = α0 , β 0 , ζ 0


0

with

(8.222)

α = (α0 , α1 , α2 , . . . , αN −1 , α11 , α12 , . . . , α1,N −1 , α22 , . . . , α2,N −1 , . . . , αN −1,N −1 )0

(8.223)

β = (β1 , β2 , . . . , βM , β11 , β12 , . . . , β1M , β22 , . . . , β2M , . . . , βM M )0

(8.224)

ζ = (ζ11 , ζ12 , . . . , ζ1M , ζ21 , ζ22 , . . . , ζ2M , . . . , ζN −1,1 , ζN −1,2 , . . . , ζN −1,M , )0

(8.225)

is the vector of estimated coefficients, and r = (0, 0, . . . , 0, −1, 0, 0, . . . , 0) is the vector of righthand-side values. If one has N inputs and M outputs and one wants to impose monotonicity at n observations, the restriction matrix R must have n·(N +M ) rows and 1+(N −1)(1+N/2)+M (3+M )/2+(N −1)M columns, where the number of columns is equal to the number of estimated coefficients. As our data set has 140 observations and we have 3 inputs and 2 outputs, our restriction matrix must have 140 · (3 + 2) = 700 rows and 1 + 2 · 2.5 + 2 · 5/2 + 2 · 2 = 15 columns, The following code generates the restriction matrix and the restriction vector:

346

8 Distance Functions > RMat colnames( RMat ) rVec # regarding capital input > rowsCap RMat[ rowsCap, "log(qCap/qMat)" ] RMat[ rowsCap, "I(0.5 * log(qCap/qMat)^2)" ] RMat[ rowsCap, "I(log(qCap/qMat) * log(qLab/qMat))" ] RMat[ rowsCap, "I(log(qCap/qMat) * log(qApples))" ] RMat[ rowsCap, "I(log(qCap/qMat) * log(qOtherOut))" ] # regarding labor input > rowsLab RMat[ rowsLab, "log(qLab/qMat)" ] RMat[ rowsLab, "I(0.5 * log(qLab/qMat)^2)" ] RMat[ rowsLab, "I(log(qCap/qMat) * log(qLab/qMat))" ] RMat[ rowsLab, "I(log(qLab/qMat) * log(qApples))" ] RMat[ rowsLab, "I(log(qLab/qMat) * log(qOtherOut))" ] # regarding materials input > rowsMat RMat[ rowsMat, "log(qCap/qMat)" ] RMat[ rowsMat, "log(qLab/qMat)" ] RMat[ rowsMat, "I(0.5 * log(qCap/qMat)^2)" ] RMat[ rowsMat, "I(0.5 * log(qLab/qMat)^2)" ] RMat[ rowsMat, "I(log(qCap/qMat) * log(qLab/qMat))" ] RMat[ rowsMat, "I(log(qCap/qMat) * log(qApples))" ] RMat[ rowsMat, "I(log(qCap/qMat) * log(qOtherOut))" ] RMat[ rowsMat, "I(log(qLab/qMat) * log(qApples))" ] RMat[ rowsMat, "I(log(qLab/qMat) * log(qOtherOut))" ] rVec[ rowsMat ] # regarding apple output > rowsApp RMat[ rowsApp, "log(qApples)" ] RMat[ rowsApp, "I(0.5 * log(qApples)^2)" ] RMat[ rowsApp, "I(log(qApples) * log(qOtherOut))" ] RMat[ rowsApp, "I(log(qCap/qMat) * log(qApples))" ] RMat[ rowsApp, "I(log(qLab/qMat) * log(qApples))" ] # regarding other outputs > rowsOth RMat[ rowsOth, "log(qOtherOut)" ] RMat[ rowsOth, "I(log(qApples) * log(qOtherOut))" ] RMat[ rowsOth, "I(0.5 * log(qOtherOut)^2)" ] RMat[ rowsOth, "I(log(qCap/qMat) * log(qOtherOut))" ] RMat[ rowsOth, "I(log(qLab/qMat) * log(qOtherOut))" ] all.equal( c( RMat %*% coef( idfTL )[1:15] - rVec ), +

c( dat$eCapTLidf, dat$eLabTLidf, dat$eMatTLidf,

+

-dat$eApplesTLidf, -dat$eOtherTLidf ) )

[1] TRUE We can obtain restricted coefficients as suggested by Henningsen and Henning (2009): > uCoef uCovInv library( "quadprog" ) > minDistResult rCoef elaIdfTl dat$eCapTLRidf dat$eLabTLRidf dat$eMatTLRidf dat$eApplesTLRidf dat$eOtherTLRidf summary( dat[ , c( "eCapTLRidf", "eLabTLRidf", "eMatTLRidf", +

"eApplesTLRidf", "eOtherTLRidf" ) ] ) eCapTLRidf

Min.

:0.00000

eLabTLRidf Min.

:0.5105

eMatTLRidf Min.

:0.1150

eApplesTLRidf Min.

:-0.2958

1st Qu.:0.06868

1st Qu.:0.6228

1st Qu.:0.1987

1st Qu.:-0.2229

Median :0.10036

Median :0.6596

Median :0.2434

Median :-0.1943

Mean

Mean

Mean

Mean

:0.09914

3rd Qu.:0.13391

:0.6571

3rd Qu.:0.6950

:0.2437

3rd Qu.:0.2737

348

:-0.1864

3rd Qu.:-0.1593

8 Distance Functions Max.

:0.20573

Max.

:0.7971

Max.

:0.4360

Max.

: 0.0000

eOtherTLRidf Min.

:-0.19191

1st Qu.:-0.15044 Median :-0.13416 Mean

:-0.13396

3rd Qu.:-0.11993 Max.

:-0.06537

We can see that all distance elassticities of the three are non-negative and all distance elasticities of the two outputs are non-positive. The homogeneity condition is—of course—still fulfilled: > range( dat$eCapTLRidf + dat$eLabTLRidf + dat$eMatTLRidf ) [1] 1 1 And the elasticities of scale can be calculated by: > dat$eScaleTLRidf summary( dat$eScaleTLRidf ) Min. 1st Qu. 2.050

2.741

Median 2.992

Mean 3rd Qu. 3.418

3.515

Max. 15.298

349

9 Panel Data and Technological Change Until now, we have only analyzed cross-sectional data, i.e. all observations refer to the same period of time. Hence, it was reasonable to assume that the same technology is available to all firms (observations). However, when analyzing time series data or panel data, i.e. when observations can originate from different time periods, different technologies might be available in the different time periods due to technological change. Hence, the state of the available technologies must be included as an explanatory variable in order to conduct a reasonable production analysis. Often, a time trend is used as a proxy for a gradually changing state of the available technologies. We will demonstrate how to analyze production technologies with data from different time periods by using a balanced panel data set of annual data collected from 43 smallholder rice producers in the Tarlac region of the Philippines between 1990 and 1997. We loaded this data set (riceProdPhil) in section 1.4.2. As it does not contain information about the panel structure, we created a copy of the data set (pdat) that includes information on the panel structure.

9.1 Average production functions with technological change In case of an applied production analysis with time-series data or panel data, usually the time (t) is included as additional explanatory variable in the production function: y = f (x, t).

(9.1)

This function can be used to analyze how the time (t) affects the (available) production technology. The average production technology (potentially depending on the time period) can be estimated from panel data sets by the OLS method (i.e. “pooled”) or by any of the usual panel data methods (e.g. fixed effects, random effects).

9.1.1 Cobb-Douglas production function with technological change In case of a Cobb-Douglas production function, usually a linear time trend is added to account for technological change: ln y = α0 +

X

αi ln xi + αt t

i

350

(9.2)

9 Panel Data and Technological Change Given this specification, the coefficient of the (linear) time trend can be interpreted as the rate of technological change per unit of the time variable t: ∆y ∂ ln y ∂ ln y ∂y y αt = = ≈ ∂t ∂y ∂t ∆x

(9.3)

9.1.1.1 Pooled estimation of the Cobb-Douglas production function with technological change The pooled estimation can be done by: > riceCdTime summary( riceCdTime ) Call: lm(formula = log(PROD) ~ log(AREA) + log(LABOR) + log(NPK) + mYear, data = riceProdPhil) Residuals: Min

1Q

Median

3Q

Max

-1.83351 -0.16006

0.05329

0.22110

0.86745

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -1.665096

0.248509

-6.700 8.68e-11 ***

log(AREA)

0.333214

0.062403

5.340 1.71e-07 ***

log(LABOR)

0.395573

0.066421

5.956 6.48e-09 ***

log(NPK)

0.270847

0.041027

6.602 1.57e-10 ***

mYear

0.010090

0.008007

1.260

0.208

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3299 on 339 degrees of freedom Multiple R-squared:

0.86,

Adjusted R-squared:

F-statistic: 520.6 on 4 and 339 DF,

0.8583

p-value: < 2.2e-16

The estimation result indicates an annual rate of technical change of approximately 1%, but this is not statistically different from 0%, i.e., we cannot reject that there has been no technological change. The output elasticities are equal to the coefficients of the (logarithmic) input quantities and the elasticity of scale is almost exactly one (0.99963), indicating (approximately) constant returns to scale.

351

9 Panel Data and Technological Change The command above can be simplified by using the pre-calculated logarithmic (and meanscaled) quantities: > riceCdTimeS summary( riceCdTimeS ) Call: lm(formula = lProd ~ lArea + lLabor + lNpk + mYear, data = riceProdPhil) Residuals: Min

1Q

Median

3Q

Max

-1.83351 -0.16006

0.05329

0.22110

0.86745

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept) -0.015590

0.019325

-0.807

0.420

lArea

0.333214

0.062403

5.340 1.71e-07 ***

lLabor

0.395573

0.066421

5.956 6.48e-09 ***

lNpk

0.270847

0.041027

6.602 1.57e-10 ***

mYear

0.010090

0.008007

1.260

0.208

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3299 on 339 degrees of freedom Multiple R-squared:

0.86,

Adjusted R-squared:

F-statistic: 520.6 on 4 and 339 DF,

0.8583

p-value: < 2.2e-16

The intercept has changed because of the mean-scaling of the input and output quantities but all slope parameters are unaffected by using the pre-calculated logarithmic (and mean-scaled) quantities: > all.equal( coef( riceCdTime )[-1], coef( riceCdTimeS )[-1], +

check.attributes = FALSE )

[1] TRUE 9.1.1.2 Panel data estimations of the Cobb-Douglas production function with technological change The panel data estimation with fixed individual effects can be done by: > riceCdTimeFe summary( riceCdTimeFe )

352

9 Panel Data and Technological Change Oneway (individual) effect Within Model Call: plm(formula = lProd ~ lArea + lLabor + lNpk + mYear, data = pdat) Balanced Panel: n = 43, T = 8, N = 344 Residuals: Min.

1st Qu.

Median

3rd Qu.

Max.

-1.585363 -0.156896

0.045606

0.177983

0.818168

Coefficients: Estimate Std. Error t-value lArea

Pr(>|t|)

0.5607756

0.0785370

7.1403 7.195e-12 ***

lLabor 0.2549108

0.0690631

3.6910 0.0002657 ***

lNpk

0.1748528

0.0484684

3.6076 0.0003625 ***

mYear

0.0130908

0.0071824

1.8226 0.0693667 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Total Sum of Squares:

43.632

Residual Sum of Squares: 24.872 R-Squared:

0.42995

Adj. R-Squared: 0.34165 F-statistic: 56.0008 on 4 and 297 DF, p-value: < 2.22e-16 The estimation result indicates an annual rate of technical change of around 1.3%, which is statistically different from 0% (only) at the 10% significance level. While the elasticity of scale is still approximately one (1.00363), the estimated output elasticities notably differ from those obtained from the pooled regression. For instance, the output elasticity of land obtained from the pooled regression is 0.33 and obtained from the fixed-effects regression is 0.56. A potential reason for these differences is an endogeneity bias in the pooled regression: if there are unoserved time-invariant variables that affect the output quantity and are correlated with some of the input quantities (e.g., the average soil quality at each farm is likely time-invariant and positively correlated with the error term and may also be positively correlated with the quantity of fertilizers), the pooled regression gives biased estimates, while the fixed effect estimation gives unbiased results, because the effects of the unoserved time-invariant variables are absorbed in the fixed effects. The panel data estimation with random individual effects can be done by: > riceCdTimeRan summary( riceCdTimeRan ) Oneway (individual) effect Random Effect Model (Swamy-Arora's transformation) Call: plm(formula = lProd ~ lArea + lLabor + lNpk + mYear, data = pdat, model = "random") Balanced Panel: n = 43, T = 8, N = 344 Effects: var std.dev share idiosyncratic 0.08375 0.28939

0.8

individual

0.2

0.02088 0.14451

theta: 0.4222 Residuals: Min.

1st Qu.

Median

3rd Qu.

Max.

-1.749676 -0.142852

0.048474

0.190936

0.851886

Coefficients: Estimate Std. Error z-value

Pr(>|z|)

(Intercept) -0.0213044

0.0292268 -0.7289

0.4660

lArea

0.4563002

0.0662979

6.8826 5.878e-12 ***

lLabor

0.3190041

0.0647524

4.9265 8.371e-07 ***

lNpk

0.2268399

0.0426651

5.3168 1.056e-07 ***

mYear

0.0115453

0.0071921

1.6053

0.1084

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Total Sum of Squares:

117.05

Residual Sum of Squares: 29.195 R-Squared:

0.75058

Adj. R-Squared: 0.74764 Chisq: 1020.18 on 4 DF, p-value: < 2.22e-16 A variable-coefficient model for panel model with individual-specific coefficients can be estimated by:

354

9 Panel Data and Technological Change > riceCdTimeVc summary( riceCdTimeVc ) Oneway (individual) effect No-pooling model Call: pvcm(formula = lProd ~ lArea + lLabor + lNpk + mYear, data = pdat) Balanced Panel: n = 43, T = 8, N = 344 Residuals: Min.

1st Qu.

Median

3rd Qu.

Max.

-0.817465311 -0.081971866

0.006676688

0.093984652

0.554124266

Coefficients: (Intercept) Min.

:-3.8110

lArea Min.

:-5.2850

lLabor Min.

:-2.72761

lNpk Min.

:-1.3094

1st Qu.:-0.3006

1st Qu.:-0.4200

1st Qu.:-0.30989

1st Qu.:-0.1867

Median : 0.1145

Median : 0.6978

Median : 0.08778

Median : 0.1050

Mean

Mean

Mean

Mean

: 0.1839

: 0.5896

: 0.06079

: 0.1265

3rd Qu.: 0.5617

3rd Qu.: 1.8914

3rd Qu.: 0.61479

3rd Qu.: 0.3808

Max.

Max.

: 4.7633

Max.

Max.

NA's

:18

: 3.7270

: 1.75595

: 1.7180

mYear Min.

:-0.471049

1st Qu.:-0.044359 Median :-0.008111 Mean

:-0.012327

3rd Qu.: 0.054743 Max.

: 0.275875

Total Sum of Squares: 2861.8 Residual Sum of Squares: 8.9734 Multiple R-Squared: 0.99686 A pooled estimation can also be done by > riceCdTimePool all.equal( coef( riceCdTimeS ), coef( riceCdTimePool ) ) [1] TRUE A Hausman test can be used to check the consistency of the random-effects estimator: > phtest( riceCdTimeRan, riceCdTimeFe ) Hausman Test data:

lProd ~ lArea + lLabor + lNpk + mYear

chisq = 14.62, df = 4, p-value = 0.005557 alternative hypothesis: one model is inconsistent The Hausman test clearly shows that the random-effects estimator is inconsistent (due to correlation between the individual effects and the explanatory variables). Now, we test the poolability of the model: > pooltest( riceCdTimePool, riceCdTimeFe ) F statistic data:

lProd ~ lArea + lLabor + lNpk + mYear

F = 3.4175, df1 = 42, df2 = 297, p-value = 4.038e-10 alternative hypothesis: unstability > pooltest( riceCdTimePool, riceCdTimeVc ) F statistic data:

lProd ~ lArea + lLabor + lNpk + mYear

F = 1.9113, df1 = 210, df2 = 129, p-value = 4.022e-05 alternative hypothesis: unstability > pooltest( riceCdTimeFe, riceCdTimeVc ) F statistic data:

lProd ~ lArea + lLabor + lNpk + mYear

F = 1.3605, df1 = 168, df2 = 129, p-value = 0.03339 alternative hypothesis: unstability The pooled model (riceCdTimePool) is clearly rejected in favor of the model with fixed individual effects (riceCdTimeFe) and the variable-coefficient model (riceCdTimeVc). The model with fixed individual effects (riceCdTimeFe) is rejected in favor of the variable-coefficient model (riceCdTimeVc) at 5% significance level but not at 1% significance level.

356

9 Panel Data and Technological Change

9.1.2 Translog production function with constant and neutral technological change A Translog production function that accounts for constant and neutral (unbiased) technological change has following specification: ln y = α0 +

X

αi ln xi +

i

1 XX αij ln xi ln xj + αt t 2 i j

(9.4)

In this specification, the rate of technological change is ∂ ln y = αt ∂t

(9.5)

and the output elasticities are the same as in the time-invariant Translog production function (2.134): i =

X ∂ ln y = αi + αij ln xj ∂ ln xi j

(9.6)

In order to be able to interpret the first-order coefficients of the (logarithmic) input quantities (αi ) as output elasticities (i ) at the sample mean, we use the mean-scaled input quantities. We also use the mean-scaled output quantity in order to use the same variables as Coelli et al. (2005, p. 250). 9.1.2.1 Pooled estimation of the Translog production function with constant and neutral technological change The following command estimates a Translog production function that can account for constant and neutral technical change: > riceTlTime summary( riceTlTime ) Call: lm(formula = lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear, data = riceProdPhil) Residuals: Min

1Q

Median

3Q

Max

-1.52184 -0.18121

0.04356

0.22298

0.87019

357

9 Panel Data and Technological Change Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

0.013756

0.024645

0.558

lArea

0.588097

0.085162

6.906 2.54e-11 ***

lLabor

0.191764

0.080876

2.371

0.01831 *

lNpk

0.197875

0.051605

3.834

0.00015 ***

-0.435547

0.247491

-1.760

0.07935 .

I(0.5 * lLabor^2) -0.742242

0.303236

-2.448

0.01489 *

I(0.5 * lNpk^2)

0.020367

0.097907

0.208

0.83534

I(lArea * lLabor)

0.678647

0.216594

3.133

0.00188 **

I(lArea * lNpk)

0.063920

0.145613

0.439

0.66097

I(lLabor * lNpk)

-0.178286

0.138611

-1.286

0.19926

0.012682

0.007795

1.627

0.10468

I(0.5 * lArea^2)

mYear

0.57712

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3184 on 333 degrees of freedom Multiple R-squared:

0.8719,

Adjusted R-squared:

F-statistic: 226.6 on 10 and 333 DF,

0.868

p-value: < 2.2e-16

In the Translog production function that accounts for constant and neutral technological change, the monotonicity conditions are fulfilled at the sample mean and the estimated output elasticities of land, labor and fertilizer are 0.588, 0.192, and 0.198, respectively, at the sample mean. The estimated (constant) annual rate of technological progress is around 1.3%. Conduct a Wald test to test whether the Translog production function outperforms the CobbDouglas production function: > library( "lmtest" ) > waldtest( riceCdTimeS, riceTlTime ) Wald test Model 1: lProd ~ lArea + lLabor + lNpk + mYear Model 2: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear Res.Df Df 1

339

2

333

F

Pr(>F)

6 5.1483 4.451e-05 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

358

9 Panel Data and Technological Change The Cobb-Douglas specification is clearly rejected in favor of the Translog specification for the pooled estimation. 9.1.2.2 Panel-data estimations of the Translog production function with constant and neutral technological change The following command estimates a Translog production function that can account for constant and neutral technical change with fixed individual effects: > riceTlTimeFe summary( riceTlTimeFe ) Oneway (individual) effect Within Model Call: plm(formula = lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear, data = pdat, model = "within") Balanced Panel: n = 43, T = 8, N = 344 Residuals: Min.

1st Qu.

Median

3rd Qu.

Max.

-1.012473 -0.144573

0.019129

0.167687

0.745525

Coefficients: Estimate Std. Error t-value Pr(>|t|) lArea

0.5828102

0.1173298

4.9673 1.16e-06 ***

lLabor

0.0473355

0.0848594

0.5578 0.577402

lNpk

0.1211928

0.0610114

1.9864 0.047927 *

I(0.5 * lArea^2)

-0.8543901

0.2861292 -2.9860 0.003067 **

I(0.5 * lLabor^2) -0.6217163

0.2935429 -2.1180 0.035025 *

I(0.5 * lNpk^2)

0.0429446

0.0987119

0.4350 0.663849

I(lArea * lLabor)

0.5867063

0.2125686

2.7601 0.006145 **

I(lArea * lNpk)

0.1167509

0.1461380

0.7989 0.424995

I(lLabor * lNpk)

-0.2371219

mYear

0.0165309

0.1268671 -1.8691 0.062619 . 0.0069206

2.3887 0.017547 *

359

9 Panel Data and Technological Change --Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Total Sum of Squares:

43.632

Residual Sum of Squares: 21.912 R-Squared:

0.49781

Adj. R-Squared: 0.40807 F-statistic: 28.8456 on 10 and 291 DF, p-value: < 2.22e-16 And the panel data estimation with random individual effects can be done by: > riceTlTimeRan summary( riceTlTimeRan ) Oneway (individual) effect Random Effect Model (Swamy-Arora's transformation) Call: plm(formula = lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear, data = pdat, model = "random") Balanced Panel: n = 43, T = 8, N = 344 Effects: var std.dev share idiosyncratic 0.07530 0.27440

0.79

individual

0.21

0.01997 0.14130

theta: 0.434 Residuals: Min.

1st Qu.

Median

3rd Qu.

Max.

-1.393176 -0.162097

0.045567

0.184209

0.798214

Coefficients: Estimate Std. Error z-value (Intercept)

0.0213211

0.0347371

0.6138

360

Pr(>|z|) 0.539357

9 Panel Data and Technological Change lArea

0.6831045

0.0922069

7.4084 1.278e-13 ***

lLabor

0.0974523

0.0804060

1.2120

0.225511

lNpk

0.1708366

0.0546853

3.1240

0.001784 **

I(0.5 * lArea^2)

-0.4275328

0.2468086 -1.7322

0.083230 .

I(0.5 * lLabor^2) -0.6367899

0.2872825 -2.2166

0.026651 *

I(0.5 * lNpk^2)

0.0307547

0.0957745

0.3211

0.748122

I(lArea * lLabor)

0.5666863

0.2059076

2.7521

0.005921 **

I(lArea * lNpk)

0.1037657

0.1421739

0.7299

0.465481

I(lLabor * lNpk)

-0.2055786

0.1277476 -1.6093

0.107560

0.0070184

0.042752 *

mYear

0.0142202

2.0261

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Total Sum of Squares:

114.08

Residual Sum of Squares: 26.624 R-Squared:

0.76662

Adj. R-Squared: 0.75961 Chisq: 1093.86 on 10 DF, p-value: < 2.22e-16 The Translog production function cannot be estimated by a variable-coefficient model for panel model with our data set, because the number of time periods in the data set is smaller than the number of the coefficients. A pooled estimation can be done by > riceTlTimePool summary(riceTlTimePool) Pooling Model Call: plm(formula = lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear, data = pdat, model = "pooling") Balanced Panel: n = 43, T = 8, N = 344 Residuals:

361

9 Panel Data and Technological Change Min.

1st Qu.

Median

3rd Qu.

Max.

-1.521838 -0.181205

0.043555

0.222979

0.870190

Coefficients: Estimate Std. Error t-value

Pr(>|t|)

(Intercept)

0.0137557

0.0246454

0.5581 0.5771201

lArea

0.5880972

0.0851622

6.9056 2.542e-11 ***

lLabor

0.1917638

0.0808764

2.3711 0.0183052 *

lNpk

0.1978747

0.0516045

3.8344 0.0001505 ***

I(0.5 * lArea^2)

-0.4355466

0.2474913 -1.7598 0.0793520 .

I(0.5 * lLabor^2) -0.7422415

0.3032362 -2.4477 0.0148916 *

I(0.5 * lNpk^2)

0.0203673

0.0979072

0.2080 0.8353358

I(lArea * lLabor)

0.6786472

0.2165937

3.1333 0.0018822 **

I(lArea * lNpk)

0.0639200

0.1456135

0.4390 0.6609677

I(lLabor * lNpk)

-0.1782859

mYear

0.0126820

0.1386111 -1.2862 0.1992559 0.0077947

1.6270 0.1046801

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Total Sum of Squares:

263.52

Residual Sum of Squares: 33.761 R-Squared:

0.87189

Adj. R-Squared: 0.86804 F-statistic: 226.623 on 10 and 333 DF, p-value: < 2.22e-16 This gives the same estimated coefficients as the model estimated by lm: > all.equal( coef( riceTlTime ), coef( riceTlTimePool ) ) [1] TRUE A Hausman test can be used to check the consistency of the random-effects estimator: > phtest( riceTlTimeRan, riceTlTimeFe ) Hausman Test data:

lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) +

chisq = 66.071, df = 10, p-value = 2.528e-10 alternative hypothesis: one model is inconsistent The Hausman test clearly rejects the consistency of the random-effects estimator. The following command tests the poolability of the model:

362

...

9 Panel Data and Technological Change > pooltest( riceTlTimePool, riceTlTimeFe ) F statistic data:

lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) +

...

F = 3.7469, df1 = 42, df2 = 291, p-value = 1.525e-11 alternative hypothesis: unstability The pooled model (riceCdTimePool) is clearly rejected in favor of the model with fixed individual effects (riceCdTimeFe), i.e. the individual effects are statistically significant. The following commands test if the fit of Translog specification is significantly better than the fit of the Cobb-Douglas specification: > waldtest( riceCdTimeFe, riceTlTimeFe ) Wald test Model 1: lProd ~ lArea + lLabor + lNpk + mYear Model 2: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear Res.Df Df 1

297

2

291

Chisq Pr(>Chisq)

6 39.321

6.191e-07 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> waldtest( riceCdTimeRan, riceTlTimeRan ) Wald test Model 1: lProd ~ lArea + lLabor + lNpk + mYear Model 2: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear Res.Df Df 1

339

2

333

Chisq Pr(>Chisq)

6 30.077

3.8e-05 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> waldtest( riceCdTimePool, riceTlTimePool )

363

9 Panel Data and Technological Change Wald test Model 1: lProd ~ lArea + lLabor + lNpk + mYear Model 2: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear Res.Df Df Chisq Pr(>Chisq) 1

339

2

333

6 30.89

2.66e-05 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

The Cobb-Douglas functional form is rejected in favor of the Translog functional for for all three panel-specifications that we estimated above. The Wald test for the pooled model differs from the Wald test that we did in section 9.1.2.1, because waldtest by default uses a finite sample F statistic for models estimated by lm but uses a large sample Chi-squared statistic for models estimated by plm. The test statistic used by waldtest can be specified by argument test.

9.1.3 Translog production function with non-constant and non-neutral technological change Technological change is not always constant and is not always neutral (unbiased). Therefore, it might be more suitable to estimate a production function that can account for increasing or decreasing rates of technological change as well as biased (e.g. labor saving) technological change. This can be done by including a quadratic time trend and interaction terms between time and input quantities: ln y = α0 +

X i

αi ln xi +

X 1 XX 1 αij ln xi ln xj + αt t + αti t ln xi + αtt t2 2 i j 2 i

(9.7)

In this specification, the rate of technological change depends on the input quantities and the time period: X ∂ ln y = αt + αti ln xi + αtt t ∂t i

(9.8)

and the output elasticities might change over time: i =

X ∂ ln y = αi + αij ln xj + αti t. ∂ ln xi j

364

(9.9)

9 Panel Data and Technological Change 9.1.3.1 Pooled estimation of a translog production function with non-constant and non-neutral technological change The following command estimates a Translog production function that can account for nonconstant rates of technological change as well as biased technological change: > riceTlTimeNn summary( riceTlTimeNn ) Call: lm(formula = lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear + I(mYear * lArea) + I(mYear * lLabor) + I(mYear * lNpk) + I(0.5 * mYear^2), data = riceProdPhil) Residuals: Min

1Q

Median

3Q

Max

-1.54976 -0.17245

0.04623

0.21624

0.87075

Coefficients: Estimate Std. Error t value Pr(>|t|) (Intercept)

0.001255

0.031934

0.039

lArea

0.579682

0.085892

6.749 6.73e-11 ***

lLabor

0.187505

0.081359

2.305

lNpk

0.207193

0.052130

3.975 8.67e-05 ***

-0.468372

0.265363

-1.765

0.07849 .

I(0.5 * lLabor^2) -0.688940

0.308046

-2.236

0.02599 *

I(0.5 * lNpk^2)

0.055993

0.099848

0.561

0.57533

I(lArea * lLabor)

0.676833

0.223271

3.031

0.00263 **

I(lArea * lNpk)

0.082374

0.151312

0.544

0.58654

I(lLabor * lNpk)

-0.226885

0.145568

-1.559

0.12005

mYear

0.008746

0.008513

1.027

0.30497

I(mYear * lArea)

0.003482

0.028075

0.124

0.90136

I(mYear * lLabor)

0.034661

0.029480

1.176

0.24054

I(mYear * lNpk)

-0.037964

0.020355

-1.865

I(0.5 * mYear^2)

0.007611

0.007954

0.957

I(0.5 * lArea^2)

---

365

0.96867 0.02181 *

0.06305 . 0.33933

9 Panel Data and Technological Change Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Residual standard error: 0.3184 on 329 degrees of freedom Multiple R-squared:

0.8734,

Adjusted R-squared:

F-statistic: 162.2 on 14 and 329 DF,

0.868

p-value: < 2.2e-16

We conduct a Wald test to test whether the Translog production function with non-constant and non-neutral technological change outperforms the Cobb-Douglas production function and the Translog production function with constant and neutral technological change: > waldtest( riceCdTimeS, riceTlTimeNn ) Wald test Model 1: lProd ~ lArea + lLabor + lNpk + mYear Model 2: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear + I(mYear * lArea) + I(mYear * lLabor) + I(mYear * lNpk) + I(0.5 * mYear^2) Res.Df Df

F

Pr(>F)

1

339

2

329 10 3.488 0.00022 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> waldtest( riceTlTime, riceTlTimeNn ) Wald test Model 1: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear Model 2: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear + I(mYear * lArea) + I(mYear * lLabor) + I(mYear * lNpk) + I(0.5 * mYear^2) Res.Df Df 1

333

2

329

F Pr(>F)

4 0.9976 0.4089

The fit of the Translog specification with non-constant and non-neutral technological change is significantly better than the fit of the Cobb-Douglas specification but it is not significantly better than the fit of the Translog specification with constant and neutral technological change.

366

9 Panel Data and Technological Change In order to simplify the calculation of the output elasticities (with equation 9.9) and the annual rates of technological change (with equation 9.8), we create shortcuts for the estimated coefficients: > a1 a2 a3 at a11 a22 a33 att a12 table( riceProdPhil$eLabor >= 0 ) FALSE

TRUE

63

281

> table( riceProdPhil$eNpk >= 0 ) FALSE

TRUE

7

337

> riceProdPhil$monoTl = 0 & eLabor >= 0 & eNpk >= 0 ) > table( riceProdPhil$monoTl ) FALSE

TRUE

85

259

20 firms have a negative output elasticity of the land area, 63 firms have a negative output elasticity of labor, and 7 firms have a negative output elasticity of fertilizers. In total the monotonicity condition is not fulfilled at 85 out of 344 observations. Although the monotonicity conditions are fulfilled for a large part of firms in our data set, these frequent violations indicate a possible model misspecification. We can use the following command to calculate the annual rates of technological change: > riceProdPhil$tc hist( riceProdPhil$tc, 15 ) The resulting graph is shown in figure 9.2. For most observations, the annual rate of technological change was between 0% and 3%.

369

0

20

40

Frequency

9 Panel Data and Technological Change

−0.05

0.00

0.05

0.10

tc

Figure 9.2: Annual rates of technological change 9.1.3.2 Panel-data estimations of a Translog production function with non-constant and non-neutral technological change The panel data estimation with fixed individual effects can be done by: > riceTlTimeNnFe summary( riceTlTimeNnFe ) Oneway (individual) effect Within Model Call: plm(formula = lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear + I(mYear * lArea) + I(mYear * lLabor) + I(mYear * lNpk) + I(0.5 * mYear^2), data = pdat) Balanced Panel: n = 43, T = 8, N = 344 Residuals: Min.

1st Qu.

Median

3rd Qu.

Max.

-1.014803 -0.143180

0.017469

0.166852

0.749361

Coefficients: Estimate Std. Error t-value

Pr(>|t|)

lArea

0.5857359

0.1191164

4.9173 1.479e-06 ***

lLabor

0.0336966

0.0869044

0.3877

370

0.698494

9 Panel Data and Technological Change lNpk

0.1276970

I(0.5 * lArea^2)

0.0623919

2.0467

0.041599 *

-0.8588620

0.2952677 -2.9088

0.003912 **

I(0.5 * lLabor^2) -0.6154568

0.2979094 -2.0659

0.039733 *

I(0.5 * lNpk^2)

0.0673038

0.1014542

0.6634

0.507613

I(lArea * lLabor)

0.6016538

0.2164953

2.7791

0.005811 **

I(lArea * lNpk)

0.1205064

0.1549834

0.7775

0.437479

I(lLabor * lNpk)

-0.2660519

0.1353699 -1.9654

0.050336 .

mYear

0.0148796

0.0076143

1.9542

0.051654 .

I(mYear * lArea)

0.0105012

0.0270130

0.3887

0.697752

I(mYear * lLabor)

0.0230156

0.0286066

0.8046

0.421743

0.0199045 -1.4044

0.161277

0.0069948

0.403458

I(mYear * lNpk)

-0.0279542

I(0.5 * mYear^2)

0.0058526

0.8367

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Total Sum of Squares:

43.632

Residual Sum of Squares: 21.733 R-Squared:

0.50189

Adj. R-Squared: 0.40469 F-statistic: 20.6552 on 14 and 287 DF, p-value: < 2.22e-16 And the panel data estimation with random individual effects can be done by: > riceTlTimeNnRan summary( riceTlTimeNnRan ) Oneway (individual) effect Random Effect Model (Swamy-Arora's transformation) Call: plm(formula = lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear + I(mYear * lArea) + I(mYear * lLabor) + I(mYear * lNpk) + I(0.5 * mYear^2), data = pdat, model = "random") Balanced Panel: n = 43, T = 8, N = 344

371

9 Panel Data and Technological Change

Effects: var std.dev share idiosyncratic 0.07573 0.27518 0.796 individual

0.01941 0.13933 0.204

theta: 0.4275 Residuals: Min.

1st Qu.

Median

3rd Qu.

Max.

-1.392702 -0.162022

0.045553

0.180242

0.790301

Coefficients: Estimate Std. Error z-value

Pr(>|z|)

(Intercept)

0.0101183

0.0389961

0.2595

lArea

0.6809764

0.0930789

7.3161 2.552e-13 ***

lLabor

0.0865327

0.0813309

1.0640

0.287348

lNpk

0.1800677

0.0554226

3.2490

0.001158 **

I(0.5 * lArea^2)

0.795272

-0.4749163

0.2627102 -1.8078

0.070644 .

I(0.5 * lLabor^2) -0.6146891

0.2907148 -2.1144

0.034481 *

I(0.5 * lNpk^2)

0.0614961

0.0980315

0.6273

0.530456

I(lArea * lLabor)

0.5916989

0.2113078

2.8002

0.005107 **

I(lArea * lNpk)

0.1224789

0.1488815

0.8227

0.410701

I(lLabor * lNpk)

-0.2531048

0.1350400 -1.8743

0.060890 .

mYear

0.0116511

0.0077140

1.5104

0.130947

I(mYear * lArea)

0.0028675

0.0265731

0.1079

0.914068

I(mYear * lLabor)

0.0355897

0.0279156

1.2749

0.202343

I(mYear * lNpk)

-0.0344049

I(0.5 * mYear^2)

0.0069525

0.0195392 -1.7608

0.078269 .

0.0071510

0.330936

0.9722

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Total Sum of Squares:

115.71

Residual Sum of Squares: 26.417 R-Squared:

0.77169

Adj. R-Squared: 0.76198 Chisq: 1112.04 on 14 DF, p-value: < 2.22e-16 The Translog production function cannot be estimated by a variable-coefficient model for panel model with our data set, because the number of time periods in the data set is smaller than the number of the coefficients.

372

9 Panel Data and Technological Change A pooled estimation can be done by > riceTlTimeNnPool all.equal( coef( riceTlTimeNn ), coef( riceTlTimeNnPool ) ) [1] TRUE A Hausman test can be used to check the consistency of the random-effects estimator: > phtest( riceTlTimeNnRan, riceTlTimeNnFe ) Hausman Test data:

lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) +

...

chisq = 21.731, df = 14, p-value = 0.08432 alternative hypothesis: one model is inconsistent The Hausman test rejects the consistency of the random-effects estimator at the 10% significance level but it cannot reject the consistency of the random-effects estimator at the 5% significance level. The following command tests the poolability of the model: > pooltest( riceTlTimeNnPool, riceTlTimeNnFe ) F statistic data:

lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) +

...

F = 3.6544, df1 = 42, df2 = 287, p-value = 4.266e-11 alternative hypothesis: unstability The pooled model (riceCdTimePool) is clearly rejected in favor of the model with fixed individual effects (riceCdTimeFe), i.e. the individual effects are statistically significant. The following commands test if the fit of Translog specification is significantly better than the fit of the Cobb-Douglas specification: > waldtest( riceTlTimeNnFe, riceCdTimeFe )

373

9 Panel Data and Technological Change Wald test Model 1: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear + I(mYear * lArea) + I(mYear * lLabor) + I(mYear * lNpk) + I(0.5 * mYear^2) Model 2: lProd ~ lArea + lLabor + lNpk + mYear Res.Df

Df Chisq Pr(>Chisq)

1

287

2

297 -10 41.45

9.392e-06 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> waldtest( riceTlTimeNnRan, riceCdTimeRan ) Wald test Model 1: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear + I(mYear * lArea) + I(mYear * lLabor) + I(mYear * lNpk) + I(0.5 * mYear^2) Model 2: lProd ~ lArea + lLabor + lNpk + mYear Res.Df

Df

Chisq Pr(>Chisq)

1

329

2

339 -10 33.666

0.0002103 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> waldtest( riceTlTimeNnPool, riceCdTimePool ) Wald test Model 1: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear + I(mYear * lArea) + I(mYear * lLabor) + I(mYear * lNpk) + I(0.5 * mYear^2) Model 2: lProd ~ lArea + lLabor + lNpk + mYear Res.Df

Df Chisq Pr(>Chisq)

1

329

2

339 -10 34.88

0.0001309 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

374

9 Panel Data and Technological Change Finally, we test whether the fit of Translog specification with non-constant and non-neutral technological change is significantly better than the fit of Translog specification with constant and neutral technological change: > waldtest( riceTlTimeNnFe, riceTlTimeFe ) Wald test Model 1: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear + I(mYear * lArea) + I(mYear * lLabor) + I(mYear * lNpk) + I(0.5 * mYear^2) Model 2: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear Res.Df Df

Chisq Pr(>Chisq)

1

287

2

291 -4 2.3512

0.6715

> waldtest( riceTlTimeNnRan, riceTlTimeRan ) Wald test Model 1: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear + I(mYear * lArea) + I(mYear * lLabor) + I(mYear * lNpk) + I(0.5 * mYear^2) Model 2: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear Res.Df Df

Chisq Pr(>Chisq)

1

329

2

333 -4 3.6633

0.4535

> waldtest( riceTlTimeNnPool, riceTlTimePool ) Wald test Model 1: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear + I(mYear * lArea) + I(mYear * lLabor) + I(mYear * lNpk) + I(0.5 * mYear^2)

375

9 Panel Data and Technological Change Model 2: lProd ~ lArea + lLabor + lNpk + I(0.5 * lArea^2) + I(0.5 * lLabor^2) + I(0.5 * lNpk^2) + I(lArea * lLabor) + I(lArea * lNpk) + I(lLabor * lNpk) + mYear Res.Df Df

Chisq Pr(>Chisq)

1

329

2

333 -4 3.9905

0.4073

The tests indicate that the fit of Translog specification with constant and neutral technological change is not significantly worse than the fit of Translog specification with non-constant and non-neutral technological change. The difference between the Wald tests for the pooled model and the Wald test that we did in section 9.1.3.1 is explained at the end of section 9.1.2.2.

9.2 Frontier production functions with technological change The frontier production technology can be estimated by many different specifications of the stochastic frontier model. We will focus on four specifications that are all nested in the general specification: ln ykt = ln f (xkt , t, k) − ukt + vkt ,

(9.10)

where the subscript k = 1, . . . , K indicates the firm, t = 1, . . . , T indicates the time period, and all other variables are defined as before. We will apply the following four model specifications: 1. the same frontier for all firms, i.e., f (xkt , t, k) = f (xkt , t) ∀ k, t, and time-invariant individual efficiencies, i.e., ukt = uk ∀ k, t, which means that each firm has an individual fixed efficiency that remains constant over time; 2. the same frontier for all firms, i.e., f (xkt , t, k) = f (xkt , t) ∀ k, t, and time-variant individual efficiencies, with ukt = uk exp(−η (t − T )) ∀ k, t, which means that each firm has an individual efficiency and the inefficiency terms ukt of all firms can change over time with the same rate (and in the same direction) as indicated by the additional coefficient η; 3. the same frontier for all firms, i.e., f (xkt , t, k) = f (xkt , t) ∀ k, t, and observation-specific efficiencies, i.e., no restrictions on ukt , which means that the efficiency term of each observation is estimated independently from the other efficiencies of the firm so that basically the panel structure of the data is ignored; and 4. a different frontier for each firm, i.e., f (xkt , t, k) = f (xkt , t) eδk ∀ k, t (i.e., individual fixed effects in the frontier), and observation-specific efficiencies, i.e., no restrictions on ukt .

9.2.1 Cobb-Douglas production frontier with technological change We will use the specification in equation (9.2).

376

9 Panel Data and Technological Change 9.2.1.1 Time-invariant individual efficiencies We start with estimating a Cobb-Douglas production frontier with time-invariant individual efficiencies. The following commands estimate two Cobb-Douglas production frontiers with timeinvariant individual efficiencies, the first does not account for technological change, while the second does: > riceCdSfaInv summary( riceCdSfaInv ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 9 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept) 0.182636

0.035340

5.1680 2.366e-07 ***

lArea

0.453900

0.064382

7.0501 1.788e-12 ***

lLabor

0.288922

0.063853

4.5248 6.045e-06 ***

lNpk

0.227542

0.040644

5.5983 2.164e-08 ***

sigmaSq

0.155377

0.024144

6.4354 1.232e-10 ***

gamma

0.464317

0.088270

5.2602 1.439e-07 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -86.43042 panel data number of cross-sections = 43 number of time periods = 8 total number of observations = 344 thus there are 0 observations not in the panel mean efficiency: 0.8187935 > riceCdTimeSfaInv summary( riceCdTimeSfaInv ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function)

377

9 Panel Data and Technological Change The dependent variable is logged Iterative ML estimation terminated after 11 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept) 0.1832751

0.0345895

5.2986 1.167e-07 ***

lArea

0.4625174

0.0644245

7.1792 7.011e-13 ***

lLabor

0.3029415

0.0641323

4.7237 2.316e-06 ***

lNpk

0.2098907

0.0418709

5.0128 5.364e-07 ***

mYear

0.0116003

0.0071758

1.6166

sigmaSq

0.1556806

0.0242951

6.4079 1.475e-10 ***

gamma

0.4706143

0.0869549

5.4122 6.227e-08 ***

0.106

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -85.0743 panel data number of cross-sections = 43 number of time periods = 8 total number of observations = 344 thus there are 0 observations not in the panel mean efficiency: 0.8176333 In the Cobb-Douglas production frontier that accounts for technological change, the monotonicity conditions are globally fulfilled and the (constant) output elasticities of land, labor and fertilizer are 0.463, 0.303, and 0.21, respectively. The estimated (constant) annual rate of technological progress is around 1.2%. However, both the t-test for the coefficient of the time trend and a likelihood ratio test give rise to doubts whether the production technology indeed changes over time (P-values around 10%): > lrtest( riceCdTimeSfaInv, riceCdSfaInv ) Likelihood ratio test Model 1: riceCdTimeSfaInv Model 2: riceCdSfaInv #Df 1

LogLik Df

Chisq Pr(>Chisq)

7 -85.074

378

9 Panel Data and Technological Change 2

6 -86.430 -1 2.7122

0.09958 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Further likelihood ratio tests show that OLS models are clearly rejected in favor of the corresponding stochastic frontier models (no matter whether the production frontier accounts for technological change or not): > lrtest( riceCdSfaInv ) Likelihood ratio test Model 1: OLS (no inefficiency) Model 2: Error Components Frontier (ECF) #Df

LogLik Df

1

5 -104.91

2

6

-86.43

Chisq Pr(>Chisq)

1 36.953

6.051e-10 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> lrtest( riceCdTimeSfaInv ) Likelihood ratio test Model 1: OLS (no inefficiency) Model 2: Error Components Frontier (ECF) #Df

LogLik Df

1

6 -104.103

2

7

-85.074

Chisq Pr(>Chisq)

1 38.057

3.434e-10 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

This model estimates only a single efficiency estimate for each of the 43 firms. Hence, the vector returned by the efficiencies method only has 43 elements by default: > length( efficiencies( riceCdSfaInv ) ) [1] 43 One can obtain the efficiency estimates for each observation by setting argument asInData equal to TRUE: > pdat$effCdInv riceCdSfaVar summary( riceCdSfaVar ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 11 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept) 0.182016

0.035251

5.1635 2.424e-07 ***

lArea

0.474919

0.066213

7.1726 7.360e-13 ***

lLabor

0.300094

0.063872

4.6983 2.623e-06 ***

lNpk

0.199461

0.042740

4.6669 3.058e-06 ***

sigmaSq

0.129957

0.021098

6.1598 7.285e-10 ***

gamma

0.369639

0.104045

3.5527 0.0003813 ***

time

0.058909

0.030863

1.9087 0.0563017 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -84.55036 panel data number of cross-sections = 43 number of time periods = 8 total number of observations = 344 thus there are 0 observations not in the panel mean efficiency of each year 1

2

3

4

5

6

7

8

0.7848433 0.7950303 0.8048362 0.8142652 0.8233226 0.8320146 0.8403483 0.8483313 mean efficiency: 0.817874

380

9 Panel Data and Technological Change > riceCdTimeSfaVar summary( riceCdTimeSfaVar ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 14 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept)

0.1817461

0.0358969

5.0630 4.127e-07 ***

lArea

0.4761159

0.0654855

7.2706 3.580e-13 ***

lLabor

0.2987926

0.0646926

4.6187 3.862e-06 ***

lNpk

0.1991403

0.0428435

4.6481 3.350e-06 ***

mYear

-0.0031916

0.0150900 -0.2115

0.83249

sigmaSq

0.1255571

0.0287422

4.3684 1.252e-05 ***

gamma

0.3478592

0.1488417

2.3371

0.01943 *

time

0.0711209

0.0657939

1.0810

0.27971

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -84.52871 panel data number of cross-sections = 43 number of time periods = 8 total number of observations = 344 thus there are 0 observations not in the panel mean efficiency of each year 1

2

3

4

5

6

7

8

0.7780268 0.7905799 0.8025751 0.8140192 0.8249213 0.8352924 0.8451453 0.8544942 mean efficiency: 0.8181318 In the Cobb-Douglas production frontier that accounts for technological change, the monotonicity conditions are globally fulfilled and the (constant) output elasticities of land, labor and fertilizer are 0.476, 0.299, and 0.199, respectively. The estimated (constant) annual rate of technological change is around -0.3%, which indicates technological regress. However, the t-test for the

381

9 Panel Data and Technological Change coefficient of the time trend and a likelihood ratio test indicate that the production technology (frontier) does not change over time, i.e. there is neither technological regress nor technological progress: > lrtest( riceCdTimeSfaVar, riceCdSfaVar ) Likelihood ratio test Model 1: riceCdTimeSfaVar Model 2: riceCdSfaVar #Df

LogLik Df

Chisq Pr(>Chisq)

1

8 -84.529

2

7 -84.550 -1 0.0433

0.8352

A positive sign of the coefficient η (named time) indicates that efficiency is increasing over time. However, in the model without technological change, the t-test for the coefficient η and the corresponding likelihood ratio test indicate that the effect of time on the efficiencies only is significant at the 10% level: > lrtest( riceCdSfaInv, riceCdSfaVar ) Likelihood ratio test Model 1: riceCdSfaInv Model 2: riceCdSfaVar #Df LogLik Df 1

6 -86.43

2

7 -84.55

Chisq Pr(>Chisq)

1 3.7601

0.05249 .

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

In the model that accounts for technological change, the t-test for the coefficient η and the corresponding likelihood ratio test indicate that the efficiencies do not change over time: > lrtest( riceCdTimeSfaInv, riceCdTimeSfaVar ) Likelihood ratio test Model 1: riceCdTimeSfaInv Model 2: riceCdTimeSfaVar #Df

LogLik Df

1

7 -85.074

2

8 -84.529

Chisq Pr(>Chisq)

1 1.0912

0.2962

382

9 Panel Data and Technological Change Finally, we can use a likelihood ratio test to simultaneously test whether the technology and the technical efficiencies change over time: > lrtest( riceCdSfaInv, riceCdTimeSfaVar ) Likelihood ratio test Model 1: riceCdSfaInv Model 2: riceCdTimeSfaVar #Df

LogLik Df

1

6 -86.430

2

8 -84.529

Chisq Pr(>Chisq)

2 3.8034

0.1493

All together, these tests indicate that there is no significant technological change, while it remains unclear whether the technical efficiencies significantly change over time. In econometric estimations of frontier models, where one variable (e.g. time) can affect both the frontier and the efficiency, the two effects of this variable can often be hardly separated, because the corresponding parameters can be simultaneous adjusted with only marginally reducing the log-likelihood value. This can be checked by taking a look at the correlation matrix of the estimated parameters: > round( cov2cor( vcov( riceCdTimeSfaVar ) ), 2 ) (Intercept) lArea lLabor

lNpk mYear sigmaSq gamma

(Intercept)

1.00

0.17

-0.12

0.04

0.43

0.47 -0.17

lArea

0.17

1.00

-0.68 -0.38 -0.08

0.02

0.05

lLabor

-0.12 -0.68

0.02

time

1.00 -0.28

0.10

0.09

-0.06 -0.08 -0.01

lNpk

0.02 -0.38

-0.28

1.00

0.02

0.03

0.01 -0.12

mYear

0.04 -0.08

0.10

0.02

1.00

0.70

0.69 -0.87

sigmaSq

0.43

0.02

-0.06

0.03

0.70

1.00

0.94 -0.85

gamma

0.47

0.05

-0.08

0.01

0.69

0.94

1.00 -0.84

time

-0.17

0.09

-0.01 -0.12 -0.87

-0.85 -0.84

1.00

The estimate of the parameter for technological change (mYear) is highly correlated with the estimate of the parameter that indicates the change of the efficiencies (time). Again, further likelihood ratio tests show that OLS models are clearly rejected in favor of the corresponding stochastic frontier models: > lrtest( riceCdSfaVar ) Likelihood ratio test Model 1: OLS (no inefficiency)

383

9 Panel Data and Technological Change Model 2: Error Components Frontier (ECF) #Df

LogLik Df

1

5 -104.91

2

7

-84.55

Chisq Pr(>Chisq)

2 40.713

4.489e-10 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> lrtest( riceCdTimeSfaVar ) Likelihood ratio test Model 1: OLS (no inefficiency) Model 2: Error Components Frontier (ECF) #Df

LogLik Df

1

6 -104.103

2

8

-84.529

Chisq Pr(>Chisq)

2 39.149

9.85e-10 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

In case of time-variant efficiencies, the efficiencies method returns a matrix, where each row corresponds to one of the 43 firms and each column corresponds to one of the 0 time periods: > dim( efficiencies( riceCdSfaVar ) ) [1] 43

8

One can obtain a vector of efficiency estimates for each observation by setting argument asInData equal to TRUE: > pdat$effCdVar riceCdSfa summary( riceCdSfa ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged

384

9 Panel Data and Technological Change Iterative ML estimation terminated after 9 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept) 0.333747

0.024468 13.6400 < 2.2e-16 ***

lArea

0.355511

0.060125

5.9128 3.363e-09 ***

lLabor

0.333302

0.063026

5.2883 1.234e-07 ***

lNpk

0.271277

0.035364

7.6709 1.708e-14 ***

sigmaSq

0.238627

0.025941

9.1987 < 2.2e-16 ***

gamma

0.885382

0.033524 26.4103 < 2.2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -86.20268 cross-sectional data total number of observations = 344 mean efficiency: 0.7229764 > riceCdTimeSfa summary( riceCdTimeSfa ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 10 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept) 0.3375352

0.0240787 14.0180 < 2.2e-16 ***

lArea

0.3557511

0.0596403

5.9649 2.447e-09 ***

lLabor

0.3507357

0.0631077

5.5577 2.733e-08 ***

lNpk

0.2565321

0.0351012

7.3083 2.704e-13 ***

mYear

0.0148902

0.0068853

2.1626

sigmaSq

0.2418364

0.0259495

9.3195 < 2.2e-16 ***

385

0.03057 *

9 Panel Data and Technological Change gamma

0.8979766

0.0304374 29.5024 < 2.2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -83.76704 cross-sectional data total number of observations = 344 mean efficiency: 0.7201094 Please note that we used the data set riceProdPhil for these estimations, because the panel structure should be ignored in these specifications and the data set riceProdPhil does not include information on the panel structure. In the Cobb-Douglas production frontier that accounts for technological change, the monotonicity conditions are globally fulfilled and the (constant) output elasticities of land, labor and fertilizer are 0.356, 0.351, and 0.257, respectively. The estimated (constant) annual rate of technological change is around 1.5%. A likelihood ratio test confirms the t-test for the coefficient of the time trend, i.e. the production technology significantly changes over time: > lrtest( riceCdTimeSfa, riceCdSfa ) Likelihood ratio test Model 1: riceCdTimeSfa Model 2: riceCdSfa #Df

LogLik Df

Chisq Pr(>Chisq)

1

7 -83.767

2

6 -86.203 -1 4.8713

0.02731 *

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

9.2.1.4 Fixed effects in the production frontier Finally, we estimate a Cobb-Douglas production frontier with individual fixed effects in the production frontier and observation-specific efficiencies. The following command estimates a CobbDouglas production frontier with fixed individual effects (not taking into account technological change: > riceCdSfaFe summary( riceCdSfaFe )

386

9 Panel Data and Technological Change Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 16 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate

Std. Error z value

Pr(>|z|)

(Intercept)

0.15387863

0.14069324

1.0937

0.274079

lArea

0.47233861

0.13314926

3.5474

0.000389 ***

lLabor

0.21814852

0.20500184

1.0641

0.287270

lNpk

0.15996035

0.11180783

1.4307

0.152524

factor(FMERCODE)2

0.38666552

0.94482481

0.4092

0.682359

factor(FMERCODE)3

0.06006811

0.94431233

0.0636

0.949280

factor(FMERCODE)4

0.28047324

0.92834233

0.3021

0.762559

factor(FMERCODE)5

0.10347919

0.88469352

0.1170

0.906887

factor(FMERCODE)6

-0.12885816

0.93300694 -0.1381

0.890153

factor(FMERCODE)7

0.26414148

0.94820476

0.2786

0.780575

factor(FMERCODE)8

0.05056796

0.93695106

0.0540

0.956958

factor(FMERCODE)9

0.10850284

0.94108468

0.1153

0.908211

factor(FMERCODE)10

0.19886092

0.95589067

0.2080

0.835200

factor(FMERCODE)11 -0.25014169

0.63033186 -0.3968

0.691485

factor(FMERCODE)12

0.83804960

0.2491

0.803270

factor(FMERCODE)13 -0.04603965

0.91817316 -0.0501

0.960009

factor(FMERCODE)14

0.97365044

0.0583

0.953524

factor(FMERCODE)15 -0.32714010

0.92265393 -0.3546

0.722916

factor(FMERCODE)16

0.00080999

0.87478739

0.0009

0.999261

factor(FMERCODE)17

0.40772295

0.84550955

0.4822

0.629649

factor(FMERCODE)18

0.39051951

0.79343994

0.4922

0.622588

factor(FMERCODE)19

0.37686324

0.81760639

0.4609

0.644845

factor(FMERCODE)20

0.24190587

0.93457208

0.2588

0.795758

factor(FMERCODE)21

0.18704412

0.92687142

0.2018

0.840072

factor(FMERCODE)22

0.13381531

0.79596839

0.1681

0.866492

factor(FMERCODE)23

0.07095477

0.92004850

0.0771

0.938528

factor(FMERCODE)24

0.04113838

0.91898711

0.0448

0.964295

factor(FMERCODE)25

0.25307508

0.92742348

0.2729

0.784946

factor(FMERCODE)26

0.05766269

0.94544161

0.0610

0.951367

factor(FMERCODE)27

0.11471950

0.96752114

0.1186

0.905616

0.20877250 0.05674580

387

9 Panel Data and Technological Change factor(FMERCODE)28

0.21469079

0.91753698

0.2340

0.814996

factor(FMERCODE)29

0.03464250

0.86888015

0.0399

0.968197

factor(FMERCODE)30 -0.42673430

0.69841342 -0.6110

0.541196

factor(FMERCODE)31

0.28453713

0.94031605

0.3026

0.762197

factor(FMERCODE)32

0.52517557

0.41089775

1.2781

0.201208

factor(FMERCODE)33

0.10655964

0.97348249

0.1095

0.912836

factor(FMERCODE)34 -0.59248448

0.91660955 -0.6464

0.518029

factor(FMERCODE)35

0.25681291

0.94203109

0.2726

0.785148

factor(FMERCODE)36

0.03796698

0.79347793

0.0478

0.961837

factor(FMERCODE)37

0.25174598

0.93294408

0.2698

0.787283

factor(FMERCODE)38

0.34948385

0.95820893

0.3647

0.715316

factor(FMERCODE)39 -0.09070768

0.83054088 -0.1092

0.913032

factor(FMERCODE)40 -0.08208637

0.96734121 -0.0849

0.932375

factor(FMERCODE)41

0.23076919

0.91519353

0.2522

0.800922

factor(FMERCODE)42

0.25959721

0.93334414

0.2781

0.780907

factor(FMERCODE)43

0.17479501

0.72549382

0.2409

0.809607

sigmaSq

0.17985657

0.01777074 10.1209 < 2.2e-16 ***

gamma

0.98666690

0.04219155 23.3854 < 2.2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -7.971807 cross-sectional data total number of observations = 344 mean efficiency: 0.7395925 The (fixed) individual effects are highly statistically significant: > lrtest( riceCdSfa, riceCdSfaFe ) Likelihood ratio test Model 1: riceCdSfa Model 2: riceCdSfaFe #Df 1 2

LogLik Df

Chisq Pr(>Chisq)

6 -86.203 48

-7.972 42 156.46

4.285e-15 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

388

9 Panel Data and Technological Change The following command should estimate a Cobb-Douglas production frontier with fixed individual effects taking into account technological change but the estimation has problems to converge (at least at my computer): > riceCdSfaFeTime summary( riceCdSfaFeTime )

9.2.2 Translog production frontier with constant and neutral technological change The specification of a Translog production function that accounts for constant and neutral (unbiased) technological change is given in (9.4).1 9.2.2.1 Observation-specific efficiencies The following commands estimate a two Translog production frontiers with observation-specific efficiencies, the first does not account for technological change, while the second can account for constant and neutral technical change: > riceTlSfa summary( riceTlSfa ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 16 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate

Std. Error z value

Pr(>|z|)

(Intercept)

3.3719e-01

2.8747e-02 11.7298 < 2.2e-16 ***

log(area)

5.3429e-01

7.9139e-02

6.7513 1.466e-11 ***

log(labor)

2.0910e-01

7.4439e-02

2.8090 0.0049699 **

log(npk)

2.2145e-01

4.5141e-02

4.9057 9.309e-07 ***

I(0.5 * log(area)^2) 1

-5.1502e-01

2.0692e-01 -2.4889 0.0128124 *

We use not only mean-scaled input quantities but also the mean-scaled output quantity in order to obtain the same estimates as Coelli et al. (2005, p. 250). Please note that the order of coefficients/regressors is different in Coelli et al. (2005, p. 250): intercept, mYear, log(area), log(labor), log(npk), 0.5*log(area)^2, log(area)*log(labor), log(area)*log(npk), 0.5*log(labor)^2, log(labor)*log(npk), 0.5*log(npk)^2.

389

9 Panel Data and Technological Change I(0.5 * log(labor)^2)

-5.6134e-01

2.7039e-01 -2.0761 0.0378885 *

I(0.5 * log(npk)^2)

-7.1029e-05

9.4128e-02 -0.0008 0.9993979

I(log(area) * log(labor))

6.2604e-01

1.7284e-01

3.6221 0.0002922 ***

I(log(area) * log(npk))

8.1749e-02

1.3867e-01

0.5895 0.5555218

I(log(labor) * log(npk))

-1.5750e-01

1.4027e-01 -1.1228 0.2615321

sigmaSq

2.1856e-01

2.4990e-02

8.7458 < 2.2e-16 ***

gamma

8.6930e-01

3.9456e-02 22.0319 < 2.2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -76.95413 cross-sectional data total number of observations = 344 mean efficiency: 0.7326115 > riceTlTimeSfa summary( riceTlTimeSfa ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 17 iterations: log likelihood values and parameters of two successive iterations are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept)

0.3423626

0.0285089 12.0090 < 2.2e-16 ***

log(area)

0.5313816

0.0786313

6.7579 1.400e-11 ***

log(labor)

0.2308950

0.0744167

3.1027 0.0019174 **

log(npk)

0.2032741

0.0448189

4.5355 5.748e-06 ***

I(0.5 * log(area)^2)

-0.4758612

0.2021533 -2.3540 0.0185745 *

I(0.5 * log(labor)^2)

-0.5644708

0.2652593 -2.1280 0.0333374 *

I(0.5 * log(npk)^2)

-0.0072200

0.0923371 -0.0782 0.9376756

I(log(area) * log(labor))

0.6088402

0.1658019

3.6721 0.0002406 ***

I(log(area) * log(npk))

0.0617400

0.1383298

0.4463 0.6553627

390

9 Panel Data and Technological Change I(log(labor) * log(npk))

-0.1370538

0.1407360 -0.9738 0.3301377

mYear

0.0151111

0.0069164

2.1848 0.0289024 *

sigmaSq

0.2217092

0.0251305

8.8223 < 2.2e-16 ***

gamma

0.8835549

0.0367095 24.0688 < 2.2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -74.40992 cross-sectional data total number of observations = 344 mean efficiency: 0.7294192 In the Translog production frontier that accounts for constant and neutral technological change, the monotonicity conditions are fulfilled at the sample mean and the estimated output elasticities of land, labor and fertilizer are 0.531, 0.231, and 0.203, respectively, at the sample mean. The estimated (constant) annual rate of technological progress is around 1.5%. A likelihood ratio test confirms the t-test for the coefficient of the time trend, i.e. the production technology (frontier) significantly changes over time: > lrtest( riceTlTimeSfa, riceTlSfa ) Likelihood ratio test Model 1: riceTlTimeSfa Model 2: riceTlSfa #Df

LogLik Df

Chisq Pr(>Chisq)

1

13 -74.410

2

12 -76.954 -1 5.0884

0.02409 *

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

Two further likelihood ratio tests indicate that the Translog specification is superior to the CobbDouglas specification, no matter whether the two models allow for technological change or not. > lrtest( riceTlSfa, riceCdSfa ) Likelihood ratio test Model 1: riceTlSfa Model 2: riceCdSfa #Df

LogLik Df

Chisq Pr(>Chisq)

391

9 Panel Data and Technological Change 1

12 -76.954

2

6 -86.203 -6 18.497

0.005103 **

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> lrtest( riceTlTimeSfa, riceCdTimeSfa ) Likelihood ratio test Model 1: riceTlTimeSfa Model 2: riceCdTimeSfa #Df 1

LogLik Df

Chisq Pr(>Chisq)

13 -74.410

2

7 -83.767 -6 18.714

0.004674 **

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

9.2.3 Translog production frontier with non-constant and non-neutral technological change The specification of a Translog production function with non-Constant and non-Neutral technological change is given in (9.7). 9.2.3.1 Observation-specific efficiencies The following command estimates a Translog production frontier with observation-specific efficiencies that can account for non-constant rates of technological change as well as biased technological change: > riceTlTimeNnSfa summary( riceTlTimeNnSfa ) Error Components Frontier (see Battese & Coelli 1992) Inefficiency decreases the endogenous variable (as in a production function) The dependent variable is logged Iterative ML estimation terminated after 21 iterations: log likelihood values and parameters of two successive iterations

392

9 Panel Data and Technological Change are within the tolerance limit final maximum likelihood estimates Estimate Std. Error z value

Pr(>|z|)

(Intercept)

0.3106562

0.0314015

9.8931 < 2.2e-16 ***

log(area)

0.5126722

0.0786138

6.5214 6.965e-11 ***

log(labor)

0.2380479

0.0745020

3.1952 0.0013974 **

log(npk)

0.2151253

0.0443911

4.8461 1.259e-06 ***

I(0.5 * log(area)^2)

-0.5095014

0.2245191 -2.2693 0.0232500 *

I(0.5 * log(labor)^2)

-0.5394564

0.2627781 -2.0529 0.0400826 *

I(0.5 * log(npk)^2)

0.0212605

0.0923038

0.2303 0.8178341

I(log(area) * log(labor))

0.6132444

0.1687103

3.6349 0.0002781 ***

I(log(area) * log(npk))

0.0683940

0.1436778

0.4760 0.6340580

I(log(labor) * log(npk))

-0.1590173

0.1481054 -1.0737 0.2829677

mYear

0.0090026

0.0074634

1.2062 0.2277309

I(mYear * log(area))

0.0050527

0.0235232

0.2148 0.8299271

I(mYear * log(labor))

0.0241186

0.0254828

0.9465 0.3439117

I(mYear * log(npk))

-0.0335256

0.0176150 -1.9032 0.0570093 .

I(0.5 * mYear^2)

0.0149772

0.0069298

2.1613 0.0306744 *

sigmaSq

0.2227259

0.0243844

9.1339 < 2.2e-16 ***

gamma

0.8957679

0.0320612 27.9393 < 2.2e-16 ***

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

log likelihood value: -70.5919 cross-sectional data total number of observations = 344 mean efficiency: 0.728398 At the mean values of the input quantities and the middle of the observation period, the monotonicity conditions are fulfilled, the estimated output elasticities of land, labor and fertilizer are 0.513, 0.238, and 0.215, respectively, and the estimated annual rate of technological progress is around 0.9%. The following likelihood ratio tests compare the Translog production frontier that can account for non-constant rates of technological change as well as biased technological change with the Translog production frontier that does not account for technological change and with the Translog production frontier that only accounts for constant and neutral technological change: > lrtest( riceTlTimeNnSfa, riceTlSfa )

393

9 Panel Data and Technological Change Likelihood ratio test Model 1: riceTlTimeNnSfa Model 2: riceTlSfa #Df

LogLik Df

Chisq Pr(>Chisq)

1

17 -70.592

2

12 -76.954 -5 12.725

0.0261 *

--Signif. codes:

0 ‘***’ 0.001 ‘**’ 0.01 ‘*’ 0.05 ‘.’ 0.1 ‘ ’ 1

> lrtest( riceTlTimeNnSfa, riceTlTimeSfa ) Likelihood ratio test Model 1: riceTlTimeNnSfa Model 2: riceTlTimeSfa #Df

LogLik Df Chisq Pr(>Chisq)

1

17 -70.592

2

13 -74.410 -4 7.636

0.1059

These tests indicate that the Translog production frontier that can account for non-constant rates of technological change as well as biased technological change is superior to the Translog production frontier that does not account for any technological change but it is not significantly better than the Translog production frontier that accounts for constant and neutral technological change. Although it seems to be unnecessary to use the Translog production frontier that can account for non-constant rates of technological change as well as biased technological change, we use it in our further analysis for demonstrative purposes. The following commands create short-cuts for some of the estimated coefficients and calculate the rates of technological change at each observation: > at atArea atLabor atNpk att riceProdPhil$tc hist( riceProdPhil$tc, 20 )

394

30 0 10

Frequency

9 Panel Data and Technological Change

−0.05

0.00

0.05

0.10

technological change

Figure 9.3: Annual rates of technological change The resulting graph is shown in figure 9.3. Most individual rates of technological change are between −4% and +7%, i.e. there is technological regress at some observations, while there is strong technological progress at other observations. This wide variation of annual rates of technological change is not unusual in applied agricultural production analysis because of the stochastic nature of agricultural production.

9.2.4 Decomposition of productivity growth In the beginning of this course, we have discussed and calculated different productivity measures, of which the total factor productivity (T F P ) is a particularly important determinant of a firm’s competitiveness. During this course, we have—amongst other things—analyzed all three measures that affect a firm’s total factor productivity, i.e. ˆ the current state of the technology (T ) in the firm’s sector, which might change due to

technological change, ˆ the firm’s technical efficiency (T E), which might change if the firm’s distance to the current technology changes, and ˆ the firm’s scale efficiency (SE), which might change if the firm’s size relative to the optimal firm size changes. Hence, changes of a firm’s (or a sector’s) total factor productivity (∆T F P ) can be decomposed into technological changes (∆T ), technical efficiency changes (∆T E), and scale efficiency changes (∆SE): ∆T F P ≈ ∆T + ∆T E + ∆SE

(9.11)

This decomposition often helps to understand the reasons for improved or reduced total factor productivity and competitiveness.

395

9 Panel Data and Technological Change

9.3 Analyzing productivity growths with Data Envelopment Analysis (DEA) > library( "Benchmarking" ) We create a matrix of input quantities and a vector of output quantities: > xMat yVec xMat0 xMat1 yVec0 yVec1 c00 c01 c11 c10 dProd0 hist( dProd0 ) > dProd1 plot( dProd0, dProd1 ) > dProd hist( dProd ) Technological changes: > dTech0 dTech1 plot( dTech0, dTech1 ) > dTech hist( dTech ) Efficiency changes: > dEff hist( dEff ) Checking Malmquist decomposition: > all.equal( dProd, dTech * dEff ) [1] TRUE

396

9 Panel Data and Technological Change

9.4 Cost functions with technical change 9.4.1 Translog cost function with constant and neutral technical change The general specification of a Translog cost function with constant and neutral technical change is: ln c(w, y) = α0 + +

+

1 2

N X

αi ln wi i=1 N X N X

+ αy ln y

(9.12)

1 αij ln wi ln wj + αyy (ln y)2 2 i=1 j=1

N X

αiy ln wi ln y + αt t

i=1

with αij = αji ∀ i, j. This specification assumes that the rate of technical change: ∂ ln c(w, y) = αt ∂t

(9.13)

is constant over time and does not depend on the input prices or the output quantity. The value of ∂ ln c(w, y)/∂t multiplied by 100 indicates by how many percent ceteris paribus the total cost changes per time period due to technical change. A negative value of ∂ ln c(w, y)/∂t indicates technical progress, because this means that total cost ceteris paribus decreases over time, while a positive value of ∂ ln c(w, y)/∂t indicates technical regress, because this means that total cost— ceteris paribus—increases over time. Furthermore, this specification assumes that the optimal cost shares of the inputs: si (w, y) =

N X ∂ ln c(w, y) αij ln wj + αiy ln y = αi + ∂ ln wi j=1

∀i

(9.14)

are not affected by technical change. This means that technical change is Hicks-neutral.

9.4.2 Translog cost function with non-constant and non-neutral technical change The general specification of a Translog cost function with non-constant and non-neutral technical change is: ln c(w, y) = α0 + +

1 2

N X

αi ln wi i=1 N X N X

+ αy ln y

N X 1 αij ln wi ln wj + αyy (ln y)2 + αiy ln wi ln y 2 i=1 j=1 i=1

397

(9.15)

9 Panel Data and Technological Change

+ αt t +

N X

αti t ln wi + αty t ln y +

i=1

1 αtt t2 2

with αij = αji ∀ i, j. In this specification, the rate of technical change is: N X ∂ ln c(w, y) = αt + αti ln wi + αty ln y + αtt t ∂t i=1

(9.16)

and, hence, it can change over time (t) and it depends on the input prices (w) and the output quantity (y). The interpretation of the rate of technical change ∂ ln c(w, y)/∂t is explained in section 9.4.1. Furthermore, in this specification, the optimal cost shares of the inputs: si (w, y) =

N X ∂ ln c(w, y) = αi + αij ln wj + αiy ln y + αti t ∀ i ∂ ln wi j=1

(9.17)

are affected by technical change. For instance, if αti is strictly positive, the optimal cost share and, thus, the optimal quantity of input i will ceteris paribus increase over time so that this type of technical change is called “input i using technical change.” In contrast, if αti is strictly negative, the optimal cost share and, thus, the optimal quantity of input i will ceteris paribus decrease over time so that this type of technical change is called “input i saving technical change.”

398

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